EXTENSIONS OF UMBRAL CALCULUS II: DOUBLE DELTA OPERATORS, LEIBNIZ EXTENSIONS AND HATTORI-STONG THEOREMS FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Abstract. We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring E* with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where E* is free of additive torsion, in which context the central issues are number-theoret* *ic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions which are motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications; we also explain how the theory of Sheffer sequences fits naturally into this framework. Contents 1. Introduction 1 2. Delta operators and E-coalgebras 4 3. Double delta operators 7 4. Universal examples 11 5. Sheffer sequences 14 6. Leibniz extensions 15 7. Leibniz extensions and double delta operators 20 8. Stably penumbral polynomials and K-theory 23 9. Hattori-Stong theorems 24 10. Topological examples 28 References 31 1. Introduction In [24] we extended the basic notions of the Roman-Rota umbral calculus [30] to the setting of delta operators over a commutative graded ring of scalars. In the process we established fundamental links between umbral calculus, the theory of formal group laws and algebraic topology. In this sequel we extend these links further, developing the theory so as to relate the umbral ideas of Sheffer sequences and Leibniz delta operators with the algebraic-topological concepts of coactions, cooperations and Hattori-Stong ____________ Date: August 1997. 1991 Mathematics Subject Classification. Primary 05A40; Secondary 55N22. 1 2 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY theorems; once again, ideas from the theory of formal group laws are central to the connections. Throughout this article it is convenient to work over rings which are free of additive torsion; we save the more general case for a future paper, thereby completing a revised version of the programme begun in [24]. Traditionally, umbral calculus has been developed over fields such as the real or complex numbers, and the main effect of working over a ring is that scalars which are integers are not in general invertible. Problems of divisibility are therefore unavoidable and our programme may be succinctly described as developing the ideas introduced by Nichols and Sweedler in [22] so that they apply in such a context. In [24] we interpreted a delta operator e over a graded ring E* as a formal differential operator acting on the polynomial ring E*[x], and considered an inverse for e as the corresponding process of formal integration. Because such an inverse does not necessarily exist, we constructed a related coalgebra (E)* over which e is an isomorphism modulo its kernel of constants. In the torsion-free case we have inclusions E*[x] (E)* EQ*[x]; where EQ* = E* Q, hence the construction of (E)* is a matter of introducing sufficient divisibility in the original ring E*[x]. However, (E)* is not usually a subalgebra of EQ*[x]; when it is, the delta operator e is said to be Leibniz, since we may then exhibit a formula for the action of e on a product of polynomials. In the torsion-free case the concept of a Leibniz delta operator is equivalent to that of a formal group law [14] over the underlying ring E*, so establishing the connection with algebraic topology. The universal Leibniz delta operator corresponds to the universal formal group law and to the universal complex-oriented cohomology theory of unitary cobordism. In this paper we observe that a general delta operator e can always be considered Leibniz by enlarging the underlying ring E* to its minimal Leibniz extension E* L(E)* EQ*; and we explain how the ring L(E)* plays a key role in the analysis of the delta operator e. Once again, the passage from E* to L(E)* should be viewed as a matter of introducing further divisibility in the original ring E*. To study these extensions, we introduce the notion of a double delta operator. Restricted to the classical umbral setting this is equivalent to the idea of a Sheffer system; in our more general Leibniz case it corresponds to the concept of a strict isomorphism between two formal group laws, which arises in algebraic topology from a cohomology theory with two complex orientations. Such a cohomology theory occurs when we form the smash product of two complex-oriented spectra, which we model by constructing a general double delta operator E .F from two given delta operators E and F . We obtain inclusions E* F* (E . F )* E* F* Q; for which issues of divisibility in E* F* are again of prime importance. EXTENSIONS OF UMBRAL CALCULUS II 3 Our central result relates all the above questions of divisibility in a cont* *ext motivated by the Hattori-Stong theorem [13, 32] of algebraic topology. We denote by K = (K*; k) the graded, discrete derivative over the ring K* = Z[u; u-1]; as the notation indicates, this is also the topological example of complex K-theory. For a general delta operator E = (E*; e), we show that the double delta operator K . E is in fact closely connected to the coalgebra L(E) *. Moreover, if we embed E* in the coefficient ring (K . E)* we obtain an inclusion L(E)* (K . E)*: The classical Hattori-Stong theorem asserts that when E is the universal delta operator, the quotient of this inclusion is a torsion free abelian group (or equivalently, that the subgroup L(E)* is pure). We describe an um- bral generalisation by establishing necessary and sufficient conditions for the corresponding result to hold for a general delta operator. When these conditions are satisfied, it follows that the divisibility properties of the ex- tension E* L(E)* are equivalent to those of the double delta operator construction E* (K . E)*. Although we prove our divisibility results entirely in terms of the umbral calculus, we give a substantial application to algebraic topology by deter- mining precise criteria for a complex-oriented cohomology theory to satisfy a Hattori-Stong theorem. This result appears to be new, and confirms that umbral techniques can provide a convenient tool for organising certain types of complicated calculation in the theory of formal group laws. Since [4] and [25] are now available, we have felt free to incorporate such topological interpretations and applications as an integral part of the text, taking advantage of the improved formulation [27] in terms of the space S3 of loops on the 3-sphere. Even so, we have tried to ensure that the non- topologist will still be able to view our extensions of the umbral calculus as a coherent and viable entity. We describe now the contents of each section in more detail. In x2 we give a swift review of delta operators and the construction of the penumbral coalgebra (E)*, and in x3 we introduce double delta operators. We illustrate both these sections with topological examples. In x4 we explain the most useful instance of a double delta operator. Given delta operators e and f over rings E* and F* respectively, we construct the ring (E . F )* and an attendant double delta operator enjoying particular universal properties. In x5 we describe how the theory of double delta operators may be reinterpreted as the study of certain Sheffer sequences. In x6 we give a detailed discussion of the minimal Leibniz extension L(E) and relate it to the theory of formal group laws. In particular we discuss the construction of generators for the Lazard ring L()* and we prove a Kummer congruence for the coefficients for any Leibniz delta operator. We then investigate in x7 how the Leibniz properties of e and f over (E . F )* depend on the corresponding properties over the original rings. In the topological case when both delta operators arise from spectra E and F , the ring (E.F )* may be viewed as a model for the algebra E*(F ) and we consider under what conditions the two are isomorphic. We examine in x8 the ring (K . E)* which has many special properties, and show that (K . E)* can 4 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY be obtained from the penumbral algebra L(E) * of the Leibniz extension of E by inverting the polynomial variable. We introduce in x9 a further extension L(E)* (E)* EQ*; where the ring (E)* is defined as the smallest pure subgroup of the stable penumbral algebra which contains E*. The umbral version of the Hattori- Stong theorem asserts that L(E)* = (E)* for the delta operator E, and in Theorem 9.5 we give necessary and sufficient criteria for this to hold; they are phrased in terms of divisibility of the coefficients of e. We discuss a number of examples, both where the Hattori-Stong theorem is valid, and where it fails. Finally, in x10 we consider the topological applications of our results. In particular, we show that the Hattori-Stong theorem and various generalisations hold for any Landweber exact theory. We should warn the reader that occasionally our terminology and notation differs slightly from that introduced in [24]. We apologise for the confusion this may cause, but believe that the changes are for the better in the context of our applications. The authors are indebted to Andrew Baker, who first saw the possibili- ties of applying umbral calculus in algebraic topology, whose early versions of [4] helped stimulate the entire project, and who offered useful insight into the theory of formal group laws. Gian-Carlo Rota has supplied enthusias- tic encouragement for over a decade, whilst Peter Landweber and Volodia Vershinin both provided helpful corrections to [24] (which are implicitly in- corporated here). The second author thanks Trinity College Cambridge and the William Gordon Seggie Brown fund of the University of Edinburgh for financial support, and St. Andrews University for its hospitality. All three authors thank the London Mathematical Society for the scheme 3 grant which enabled them to meet during the drafting of this paper. 2. Delta operators and E-coalgebras In this section we give a summary of the background information that we need from [24], with embellishments provided by [27]. Throughout our work, E* will be a commutative ring with identity, graded by dimension and free of additive torsion; unless otherwise stated, a homo- morphism between two such rings will always respect the product, grading and identity. We abbreviate E* Q to EQ*. For any graded E*-module M* with augmentation, we write M* = E* fM*, where fM* is the kernel of the augmentation map; we then interpret E* as the submodule of scalars. The binomial coalgebra E*[x] over E* is the free left E*-module on generators 1, x, x2 : :,:where x is an indeterminate of dimension 2, invested with the coproduct Xn n ffi :xn 7- ! xn-i xi; n = 1; 2; : ::: i=0 i Together with this diagonal the usual product of polynomials makes E*[x] into a Hopf algebra. EXTENSIONS OF UMBRAL CALCULUS II 5 We write D for the linear operator d=dx acting on E*[x], and then a delta operator e over E* is a formal differential operator D2 Dk (2.1) e = D + e1___ + . .+.ek-1 ___+ . . . 2! k! in the divided power series ring E*{{D}}, where E-2n = E2n for all n. We will assume that ek lies in E2k. Therefore since D has dimension 2 (being dual to x) so does e. Since E*{{D}} is the graded E*-linear dual of E*[x], it too admits a Hopf algebra structure. Observe that (2.2) ffi O e = (e 1) O ffi = (1 e) O ffi as functions E*[x] ! E*[x]E*E*[x]. Note also that by dualising the divided powers (e)k=k! we obtain a new sequence of generators (2.3) Be0(x) = 1; Be1(x); Be2(x); : : : for E*[x] over E*. Under ffi, these generators also satisfy the binomial prop- erty Xn n ffi :Ben(x) 7- ! Ben-i(x) Bei(x); i=0 i and are known as the associated sequence of e. We denote the pair (E*; e) by E, and will usually refer to E itself as a delta operator; this convention merely makes explicit the ring on which e acts. All the above constructions, and also those below, may be reformulated by appealing to some straightforward results from [33], as follows. Since E*[x] is a Hopf algebra it is automatically a module over its dual, and this structure allows us to redefine e as the action of the substitution functional xk 7! ek-1. As was first understood by Rota, this functional is the basis for the traditional `representative' notation of the umbral calculus, and the interested reader may readily transcribe our work into this format at leisure. We need to make explicit the notion of a morphism fl between delta operators E = (E*; e) and F = (F*; f), where e = D + e1D2=2! + . .+. ek-1Dk=k! + . .a.nd f = D + f1D2=2! + . .+.fk-1Dk=k! + . ... Definition 2.4. A morphism fl :E ! F of delta operators is a homomor- phism of graded rings fl*: E* ! F* such that fl*(ek) = fk for all k 1. Thus fl carries e to f, by inducing a commutative diagram E*[x] --fl*-!F*[x] ? ? e ?y ?yf E*[x] --fl*-!F*[x] This is in fact a more restrictive definition than the one we originally gave in [24], where the commutativity of the diagram alone (but via a Hopf alge- bra map) was required. When F* is torsion-free, however, the two definitions coincide, so there will be no inconsistency in considering only Definition 2.4 in what follows. Our reasons for this choice are twofold: it will make for 6 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY ease of application, and it will be more convenient when we generalise to double delta operators. If E = (E*; e) is a delta operator, a left E-coalgebra is a triple (R*; ; r), where R* is a graded left augmented E*-coalgebra, : eR*! R* is an E*- isomorphism related to the coproduct in R* by the identities analogous to (2.2), and r is an E*-coalgebra map which makes the diagram eE*[x]- -r-! eR* ? ? e ?y ?y E*[x] - -r-! R* commute. There is a universal example of this concept, which in our torsion-free case may be described as ((E)*; e; i). Here (E)* EQ*[x] is the penumbral coalgebra of polynomials satisfying certain umbral integrality conditions, e is the given delta operator, and i is the inclusion E*[x] ! (E)*. As a left E*-module, (E)* is freely generated by a sequence of polynomials be0(x) = 1; be1(x); be2(x); : : : known as the normalised associated sequence of e. Under ffi, this sequence satisfies the divided power property Xn ffi :ben(x) 7- ! ben-i(x) bei(x); i=0 and is linked via i to (2.3)by the relationship i Ben(x) = n!ben(x). The ben(x) satisfy eben(x) = ben-1(x), and ben(0) = 0 for n > 0, so that e i fi e ff (2.5) ( ) fibj(x) = ffii;j; where for any operator and any polynomial f(x) we let fi ff fifi fif(x) = f(x)fi : x=0 As outlined in [27], a major source of examples of the above ideas is the theory of complex-oriented spectra in algebraic topology, for details of which we refer the reader to [2]. For such a spectrum E, the space S3 of loops on the 3-sphere has homology and cohomology modules (2.6) E*(S3) ~=E*[x] and E*(S3) ~=E*{{D}}; where x in E2(S3) is carried by the bottom cell S2 S3, and D in E2(S3) by pullback along the evaluation map SS3 ! S3. Under the cap product, D acts as d=dx, so that these modules are dual Hopf algebras of the type described above. The coproduct in E*(S3) and the product in E*(S3) arise from the diagonal map, whilst the product in E*(S3) and coproduct in E*(S3) arise from the H-space structure given by composition of loops. Antipodes are induced by reversing the loop parameter. A canonical map j :S3 ! CP 1 into infinite dimensional complex pro- jective space may be defined (up to homotopy) in several ways, the simplest being as a representative for a generator of the second integral cohomology EXTENSIONS OF UMBRAL CALCULUS II 7 group H2(S3) ~= Z. Then the given complex orientation te in E2(CP 1) pulls back to j*te in E2(S3), which by virtue of (2.6)may be expressed as D2 Dk D + e1___ + . .+.ek-1 ___+ . .;. 2! k! where each ek lies in E2k. In this way, our spectrum E and its complex orientation te give rise to a delta operator (E*; e) = (E*; j*te). The for- mula (2.2)expresses the standard interaction between cap product and di- agonal. The resulting sequence of elements Ben(x) in E*(S3) are E*-module gen- erators dual to the divided powers of e, and satisfy the binomial property. In this context, the triple ((E)*; e; i) is actually E*(CP 1); \ te; j* , since \ te is the Thom isomorphism. All relevant structures are compatible, since j commutes with diagonals. The generators ben(x) for E*(CP 1) are the usual duals [2] of the powers of te. An important topological example is that of K-theory. This provides us with the delta operator K = (Z[u; u-1]; k), where u 2 K2 and k = u-1(euD -1) is the discrete derivative. Note that connective K-theory yields the delta operator k = (Z[u]; k), while ordinary cohomology gives rise to the delta operator H = (Z; D). Finally, let us recall the universal delta operator of [24]. This is defined over the ring * = Z[OE1; OE2; : :]:with operator D2 Dk OE= D + OE1___ + . .+.OEk-1 ___+ . .:. 2! k! It is universal in the sense that there is a canonical morphism of delta operators e : ! E for any E, given by sending OEn to en. 3. Double delta operators We now introduce two central ideas, recalling our blanket assumption that all coefficient rings are free of additive torsion. Definition 3.1. A double delta operator consists of a pair of delta operators 1, 2 over a graded commutative ring G* with identity, together with an operator equation 2 = 1 + g121+ . .+.gk-1k1+ . . . = g(1); where gk lies in G2k for each k. P We write g0 = 1 and will refer to the formal power series g(y) = i0 giyi in G*[[y]] as a strict isomorphism from 1 to 2, by analogy with the nomenclature of the theory of formal group laws. ThePcompositional inverse (or reverse, or conjugate) power series __g(y) = i0 __giyi is, of course, a strict isomorphism from 2 to 1, and has coefficients which are integral combinations of those of g(y), given by __g __ __ 2 (3.2) 0= 1; g1= -g1; g2= 2g1 - g2; : ::: 8 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Since the set of delta operators over a fixed ring G* forms a group under composition of divided power series, there is always an expression of the form 21 0 k1 2 = 1 + g01__ + . .+.gk-1 ___+ . .;. 2! k! where g0k2 G2k for each k; see (4.3). Thus the thrust of Definition 3.1 is that each coefficient g0k-1of this series should be divisible by k! in G2k-2. So if G* is a field (as in the classical cases R and C), or at least a Q-algebr* *a, then any two delta operators are strictly isomorphic and define a unique double delta operator. We remark that for each double delta operator there will be two associ- ated sequences of polynomials b1n(x) and b2n(x) in GQ*[x], and that classic formulae of the umbral calculus such as n 2 -1 2 b1n(x) = x ___ x bn(x); n = 1; 2; : : : 1 (see [30, Corollary 3.8.2]) explain how to relate them. These formulae appear at first sight to involve scalars in GQ*, but the following result shows that the coefficients in fact belong to G*. Proposition 3.3. In GQ*[x], the two associated sequences are related as Xk X l b1k(x) = gm11gm22. .g.mk-1k-1b2l(x); l=1 m1; m2; : :;:mk-1 where the inner summation is over all sequences (m1; m2; : :;:mk-1) of nat- ural numbers such that m1 + 2m2 + . .+.(k - 1)mk-1 = k - l, and m1 + m2 + . .+.mk-1 l, so that the multinomial coefficient l l! = ___________________________________________ m1; m2; : :;:mk-1 m1!m2! . .m.k-1!(l - m1 - m2 - . .-.mk-1)! is defined. P Proof. OverfGQ*,iwefmayfwrite b1k(x) = ik;ib2i(x). To evaluate k;lwe __ apply l2fi to both sides, substitute 2 = g(1), and use (2.5). |__| We shall write G for the double delta operator (G*; 1; 2; g). Note that the action of forgetting one or other of the delta operators, and the isomorphism between them, associates two (single) delta operators to each double delta operator, which we write as 1G = (G*; 1) and 2G = (G*; 2). It follows from Proposition 3.3 that the penumbral coalgebras (1G)* and (2G)* are equal; we therefore denote their common value by (G)*. Examples of double delta operators are provided by cohomology theories with two given complex orientations. If t1; t2 2 E2(CP 1) are two orien- tations, then since E*(CP 1) ~= E*[[t2]] we can write t1 = g(t2) for some power series with coefficients in E*. Then (E*; j*t1; j*t2; g) is a double delta operator. EXTENSIONS OF UMBRAL CALCULUS II 9 Definition 3.4. A morphism : G ! H of double delta operators G and H is a ring homomorphism *: G* ! H* which induces a commutative dia- gram G*[x] --*-! H*[x] ? ? j ?y ?yj G*[x] --*-! H*[x] for j = 1, 2, in other words, * gives rise to morphisms j : jG ! jH, for j = 1, 2. Note that it follows that *(gk) = hk for each coefficient of the respective strict isomorphisms. Our other main idea involves extra data. Definition 3.5. Given two delta operators, e over E* and f over F*, a (e; f)-operator is a double delta operator G, together with morphisms of single delta operators : E ! 1G and ae: F ! 2G. Thus there are ring homomorphisms *: E* ! G* and ae*: F* ! G* which induce commutative diagrams E*[x] --*-! G*[x] F*[x] --ae*-!G*[x] ? ? ? ? e ?y ?y1 and f ?y ?y2 ae* E*[x] --*-! G*[x] F*[x] ---! G*[x] Where necessary to keep track of the double delta operator in question (such as in Proposition 4.10 below), we shall write and ae as G and aeG , respectively. Definition 3.6. A morphism from the (e; f)-operator (G; ; ae) to the (e0; f0)-operator (G0; 0; ae0) is a triple (l; ; r), where both l:E ! E0 and r: F ! F 0are morphisms of delta operators and : G ! G0 is a morphism of double delta operators, such that the diagrams ae E - --! 1G F - --! 2G ? ? ? ? l?y ?y1 and r ?y ?y2 0 ae0 E0 - --! 1G0 F 0- --! 2G0 commute. If l and r are identity functions, we refer to simply as a morphism of (e; f)-operators. We illustrate these concepts with an example drawn from number theory. Example 3.7. If p is prime, we define the Artin-Hasse delta operator as A = Z[v]; a , with v 2 A2p-2 and a given by the inverse relation (a)p p+1 (a)p2 pi-1+...+p+1(a)pi (3.8) D = a + v ______+ v _______+ . .+.v _______+ . .:. p p2 pi In effect we are defining a by giving its inverse in the group of delta oper- ators over A*. It is clear that an = 0 unless n is a multiple of p - 1. 10 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY We define a double delta operator G as follows. Let G* = Z(p)[u], where u 2 G2, let 1 be the image of the K-theory operator k under the inclusion k* = Z[u] G*, and let 2 be the image of the Artin-Hasse operator a under the map A* = Z[v] G* given by v 7! up-1. A result of Hasse [12] shows that 0 1 Y -(m)=m 1 = u-1 @ 1 - (u2)m - 1A : p-m This is a power series in 2 with coefficients in G*, showing that (G*; 1; 2) is a double delta operator; by construction it is a (k; a)-operator. More examples are provided by the motivating theory of complex-oriented spectra, which we now describe. We suppose given two such spectra E and F , and hence two delta operators E and F , with the additional property that the ring G* = (E ^ F )* ~=E*(F ) ~=F*(E) is free of additive torsion. This is the case, for example, when E is the Eilenberg-Mac Lane spectrum H representing integral cohomology, the spec- trum K representing complex K-theory, or the Thom spectrum MU repre- senting complex cobordism, and F is either K or MU. We have two natural inclusions, the left and right units, l*: E* ! (E ^ F )* and r*: F* ! (E ^ F )*; which give rise to two complex orientations l*(te) = tl and r*(tf) = tr in (E ^ F )2(CP 1). Thus there is an expansion tr= tl+ g1t2l+ . .+.gk-1tkl+ . . . = g(tl); where gk lies in (E ^F )2k for each k. Pulling back along j* to (E ^F )2(S3) yields delta operators land r, with r = g(l). Hence we have a double delta operator E*(F ); l; r . Moreover, it is a (e; f)-operator, since the diagrams E*(S3) - l*--!(E ^ F )*(S3) F*(S3) --r*-! (E ^ F )*(S3) ? ? ? ? e ?y ?yl and f ?y ?yr r* 3 E*(S3) - l*--!(E ^ F )*(S3) F*(S3) ---! (E ^ F )*(S ) commute by construction. Experts may prefer to interpret the above diagrams in terms of Boardman homomorphisms. If ml:E ! E0 and mr: F ! F 0are maps of complex-oriented spectra, then the triple (ml; ml^mr; mr) induces a morphism from the corresponding (e; f)-operator to the corresponding (e0; f0)-operator. As we shall see, not all double delta or (e; f)-operators, nor all their morphisms, arise in this fashion. EXTENSIONS OF UMBRAL CALCULUS II 11 4. Universal examples We now explain how to construct a universal (e; f)-operator for fixed delta operators E and F , and then describe a particular case which is also universal for all double delta operators. We make the constructions first, and then interpret the sense in which they are universal. Our initial step is to extend both e and f in the obvious fashion over the ring E* F* Q, where the tensor products are taken over Z. In this context, we write elements e f q as qe f, and abbreviate them further to qef so long as E* and F* are distinct; if E and F coincide, we shall refer to e1 as l, and to 1e as r, construing them as the left and right versions of e respectively. Applying the expansion theorem [30] for f in terms of e yields the operator equation X f = gk-1(e)k; k1 with fi ff fi gk-1 = f fibek(x) = fbek(x)fifi fi x=0 ff in E* F* Q. We may conveniently compute f fibek(x) by the umbral substitution (4.1) bek(f) fk fk-1: For example, since we may read off be1(x)= x; (4.2) be2(x)= 1_2(x2 - e1x); 3 2 2 be3(x)= 1_6x - 3e1x + (3e1 - e2)x from [24], it follows that (4.3) g0 = 1; g1 = 1_2(f1 - e1); g2 = 1_6(f2 - 3e1f1 + 3e21- e2): Definition 4.4. Let (E . F )* be the subring of E* F* Q multiplicatively generated over E*F* by the elements g1, g2; : :,:or equivalently, using (3.2), by the elements __g1, __g2; : :.: Appealing to the umbral notation again, it is convenient shorthand to write (E . F )* = E* F* = E* F*; where the angled brackets denote that the generated rings need not be poly- nomial. By construction, E . F = (E . F )*; e; f; g is a double delta operator, and along with the natural inclusions : E* ! (E . F )* and ae: F* ! (E . F )* is a (e; f)-operator. Whenever there are morphisms fl :E ! E0 and ffi :F ! F 0of delta op- erators, then there is a unique morphism fl . ffi from the (e; f)-operator E . F to the (e0; f0)-operator E0. F 0. We have made significant use of the lack of torsion in E* and F* in con- structing E . F ; we leave discussion of the corresponding object for general delta operators until the sequel. 12 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY By way of illustration of our construction, we describe how it works in some specific cases. Recall the definitions of the delta operators H, k, K and the universal defined at the end of x2. Proposition 4.5. The ring (H . E)* EQ* is generated over E* by the elements en=(n + 1)! for n 1. Proof. The coefficients of the power series expressing e in terms of D, as__ in (2.1), are en=(n + 1)!. |__| Corollary 4.6. The ring (H . k)* is the subring of Q[u] generated by the elements up-1=p, where p is prime. Proof. By Proposition 4.5, (H . k)* is generated by the elements un=(n + 1)!. It thus contains the subring generated by the elements up-1=p, but it is easy to see that the two subrings are equal, having the elements un=m(n) as an __ additive basis, where m(n) is the function of [1]. |__| __ Corollary 4.7. The ring (H . K)* is Q[u; u-1] = KQ*. |__| We consider the double delta operator K . E for general torsion-free E in much greater detail in x8. Proposition 4.8. The ring ( . )* is a polynomial algebra *[b1; b2; : :]: on infinitely many generators, where each bk has dimension 2k. Proof. We must identify the subring of * * Q generated by elements of the form OEi 1, 1 OEj and bk = bOE1k+1(1 OE), where the polynomials bOEk(x) are precisely the conjugate Bell polynomials as explained in [24]. We may therefore write k (k + 1)!bk = 1 OEk - OE1 OEk-1 + . .-.OEk 1; 2 where all the coefficients on the right are integers. Note that the ellipses are not meant here to indicate that pattern of the large number of omitted terms is obvious! Thus the elements 1 OEj are all in the subring generated by the OEi 1 and the bk; and since these latter two sequences of elements are clearly __ algebraically independent over the integers, (.)* has the stated form. |__| The algebra (.)* plays a crucial role in the theory of Sheffer sequences; see x5 and [28]. It is important to observe, as Corollaries 4.6 and 4.7 show, that the el- ements be(f) are not always algebraically independent, and therefore that (E . F )* is not necessarily polynomial. Clearly, however, it is always a quo- tient of a polynomial algebra, and has both E* and F* as subrings. We can now describe the universal properties of E . F , referring back to Definitions 3.1 and 3.5 for notation. Proposition 4.9. Any (e; f)-operator (G; ; ae) admits a canonical mor- phism : E . F ! G. EXTENSIONS OF UMBRAL CALCULUS II 13 Proof. Define the homomorphism *: E*F* ! G* by *(ef) = l(e)r(f). Extend it to all of (E . F )* by means of the formula * bek-1(f) = gk. It is a simple matter to check that this is a valid extension, and that all_the required diagrams commute. |__| The crux of this property is that (E.F )* is, in a certain sense, the simple* *st ring over which e and f can be made strictly isomorphic. We refer to E . F as the universal (e; f)-operator. As with any universal object, it is unique up to the relevant concept of isomorphism; see Definition 3.6. Let us briefly consider the relationship between E .F and F .E. These are defined respectively (and more precisely than hitherto) as the double delta operators (E . F )*; e1 ; 1f ; g and (F . E)*; f1 ; 1e ; __g, where we may take (E .F )* as E*F* and (F .E)* as F*E*. The switch map o :E* F* Q ! F* E* Q therefore restricts to an algebra isomorphism (E . F )* ! (F . E)*, and so invests (F . E)* with the structure of a (e1 ; 1f )-operator. Then (1; o; 1) is an isomorphism E . F ! F . E of (e; f)-operators. Returning to our example . of Proposition 4.8, we note that as well as being the universal (OE; OE)-operator it has one further universal property. Proposition 4.10. Any double delta operator G admits a canonical mor- phism : . ! G; moreover, if G is a (e; f)-operator, then extends to a canonical morphism (e; ; f) from the (OE; OE)-operator . . Proof. By the universality of there are morphisms of single delta operators 1 : ! 1G and 2 : ! 2G. In consequence, we may define the homomor- phism *: * * ! G* by means of the formula *(ff fi) = 1*(ff)2*(fi), for all ff and fi in *. The domain of may then be extended to ( . )* by the rule *(bk) = gk for all k. Now suppose that G is a (e; f)-operator. We have further morphisms of delta operators e : ! E and f : ! F . By definition, we have e 1 G* (OEi) = *(OEi) = (1* 2*)(OEi 1); so that G O e = O . as morphisms of delta operators. Similarly, aeG O f = O ae. . Thus (e; ; f) is indeed a morphism of the required__ type. |__| Because of this result, we may refer to . as the universal double delta operator. In fact, given arbitrary delta operators E and F , it may be used to define E . F by an alternative procedure. This process should be compared with the constructions made in [4], and relies on the fact that, by construction, ( . )* is a bimodule over *. Thus, we may form the ring E* * ( . )* * F*, where the first tensor product uses the *- module structure on E* furnished by e, and the second uses the *-module structure on F* furnished by f. Proposition 4.11. The ring E** (.)** F* is isomorphic to (E .F )* and, together with the operators e 1 1 and 1 1 f, constitutes a (e; f)-operator isomorphic to E . F . 14 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Proof. Proposition 4.10, provides a morphism . ! E . F and in turn a homomorphism ff: E* * ( . )* * F* ! E* E* (E . F )* F* F* = (E . F )*: This map is onto as all the generators of (E .F )* are hit; clearly the subrings E* and F* are in the image, but the elements be(f) are all in the image of the elements 1 bOE(OE) 1. That ff is monic is seen by considering the following commutative diagram in which the vertical maps are inclusions in the respective rationalisations. E* * ( . )* * F* - -ff-! (E . F )* ?? ? y ?y ~= E* * * * * F* Q - --! E* F* Q Finally we check that the delta operators agree. The operator e on (E .F )* is defined by the canonical map E* ! (E .F )*, but this map factors through E* ( . )* F* inducing the operator e 1 1 as claimed. Similarly __ for f. |__| 5. Sheffer sequences A double delta operator is determined by a divided power series 1 and a power series g over the graded ring G*, for the second delta operator is given by 2 = g(1). And giving g is equivalent to specifying the unit g(1)=1 in G*[[1]]. But the same ingredients give rise to the concept of a Sheffer sequence, which is centrally important in the theory of umbral calculus. We briefly explore the connection here and explain how, just as the universal (or generic) binomial sequence lies in the ring *[x], we can define the concept of a universal Sheffer sequence which lies in the ring ( . )*[x]. In fact the whole theory of Sheffer sequences may be recast in the context of double delta operators. We shall merely sketch how this is possible, and leave the interested reader to supply the details. Definition 5.1. The (normalised) Sheffer sequence associated to a double delta operator G = (G*; 1; 2; g) is the sequence of polynomials 2 1 sGn(x) = ___bn(x) 2 GQ*[x]; 1 where b1n(x) is the normalised associated sequence of the delta operator 1G. This corresponds in the terminology of [24, Definition 4.11] to the Sheffer sequence for the pair (__g; 1G). In [28] Sheffer sequences are studied mainly in terms of the unnormalised polynomials SGn(x) = n!sGn(x). Recall that for a double delta operator, we can write 2 = g(1) = 1(1 + g11 + g221+ . .).; where 1b1n(x) = b1n-1(x), so that Xn (5.2) sGn(x) = gjb1n-j(x); j=0 EXTENSIONS OF UMBRAL CALCULUS II 15 which shows that sGn(x) belongs to the penumbral coalgebra (G)*. Definition 5.3. A (normalised) Sheffer system consists of a delta opera- tor E = (E*; e) and a sequence of polynomials sn(x) 2 EQ*[x], where sn(x) has degree n and s0(x) = 1, such that esn(x) = sn-1 (x), and sn(0) 2 E2n. Since, by (5.2), sGn(0) = gn 2 E2n, a double delta operator gives rise to a Sheffer system, but clearly the process is reversible. The Sheffer system E; sn(x) determines the double delta operator (E*; e; s), where X s = sn(0)(e)n+1 : n0 The work of x4 now tells us that the concept of morphism of Sheffer systems E; sn(x) ! F; rn(x) must be a morphism fl :E ! F of delta operators such that fl* sn(x) = rn(x). Equivalently, it is a morphism of double delta operators (E*; e; s) ! (F*; f; r). The Sheffer system corresponding to the double delta operator . is the universal Sheffer system. The polynomial s.n (x) is written as sn;OE(x) in [28], where its universal properties are elaborated. Note that the vari- able k of [28] corresponds to bk in Proposition 4.8 so that the ring of that paper is isomorphic to our ( . )*. We end this section by drawing attention to Roman-Rota's formula for delta operators (see [30, Theorem 2.3.8]). Theorem 5.4. Suppose E; sn(x) is a Sheffer system and is a delta operator defined over E*. Then Xn fi ff sn(x) = fibek(x) sn-k (x): k=0 We leave the expert to interpret this too in terms of Boardman's universal (co)homology operation [7]. 6. Leibniz extensions We consider here the Leibniz property of a delta operator. Since all our delta operators are torsion-free, we are able to take a slightly different approach from that of [24]. Rather than define a delta operator (E*; e) as Leibniz when there exists a formula expressing how the operator e acts on a product (hence the name), we concentrate, dually, on the closure of the penumbral coalgebra under multiplication. This leads us to define, in the case where a delta operator E is not Leibniz, the minimal Leibniz extension L(E). It differs in general from the universal Leibniz extension LE of [24, x8]. In particular, and contrary to what was asserted in [24], LE* may have torsion when E* is torsion-free. We therefore leave discussion of the relationship between these two extensions to the sequel. Definition 6.1. A torsion-free delta operator E is Leibniz if the penumbral coalgebra (E)* is a subring of EQ*[x]. 16 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Since the polynomials ben(x) form a basis for EQ*[x] as an EQ*-module, there are always elements e(i; j; m) 2 EQ2(i+j-m) such that i+jX (6.2) bei(x)bej(x) = e(i; j; m)bem(x); m=1 for each i; j 1. The delta operator E is Leibniz precisely when all the e(i; j; m) belong to E*. It is clear therefore how to extend the ring E* in order that a torsion-free delta operator becomes Leibniz. Definition 6.3. For a torsion-free delta operator E, the minimal Leibniz extension of E is the delta operator L(E) = L(E)*; e , where L(E)* is the subring of EQ* generated over E* by the elements e(i; j; m) of (6.2). These generators incorporate an enormous amount of redundancy. We make a preliminary simplification by concentrating on the elements e(i; j; 1), which we shall abbreviate to e(i; j). They may be specified by the rational equations (6.4) e(i; j) = (beibej)(e) ek ek-1; which are to be interpreted by first multiplying together the rational poly- nomials bei(x) and bej(x), and then making the usual umbral substitution xk ek-1 for all k i + j. The order of performing these operations is important! A simple computation, appealing to (4.2), reveals the first few examples to be e(1; 1)= e1; e(1; 2) = e(2; 1)= 1_2(e2 - e21); (6.5) e(1; 3) = e(3; 1)= 1_6(e3 - 4e1e2 + 3e31); e(2; 2)= 1_4(e3 - 2e1e2 + e31): Lemma 6.6 (See [2, Part II, x3]). For each i and j, and for each 1 < m i + j, the element e(i; j; m) is an integer polynomial in the elements e(i0; j0) with i0+ j0 < i + j. Proof. We use induction on m, noting that the statement is empty for m = 1. Working in the Hopf algebra EQ*[x], we apply the diagonal to both sides of (6.2), and equate coefficients of bem-r(x) ber(x) (with 0 < r < m) to obtain X e(i; j; m) = e(i - s; j - t; m - r)e(s; t; r); where the summation is over 0 s i and 0 t j, with 0 < s + t < i + j.__ The inductive step now follows. |__| Proposition 6.7. The minimal Leibniz extension L(E)* may be obtained __ from E* by adjoining the elements e(i; j). |__| In general all the relations in L(E)* either arise from the relations amongst the ei in E*, or else follow from the integral equations i!j!e(i; j) = (BeiBej)(e) ek ek-1: EXTENSIONS OF UMBRAL CALCULUS II 17 When E is Leibniz, the multiplicative structure enjoyed by the coalge- bra (E)* makes it into a Hopf algebra. Its dual the ring E*[[e]] is then also a Hopf algebra whose diagonal is given by X (e) = e 1 + 1 e + e(i; j)(e)i (e)j: i;j>0 This series is a formal group law for which (2.1)is the exponential series. Thus a Leibniz delta operator is precisely one for which the corresponding formal group law is defined over the ring E*. In general this formal group law will be defined only over the extension L(E)*. If we invert the series (2.1), we may write (e)2 (e)k (6.8) D = e + c1______+ . .+.ck-1 ______+ . . . 2 k (note the absence of factorials), where the coefficients cn belong to L(E)* but not, in general, to E*; see, for example, [10, IV, x1, Proposition 1]. This is the logarithm series of the associated formal group law. In the case of the universal delta operator , the elements cn can be expressed as integer polynomials in the generators OEk, where cn -n!OEn mod decomposables in * : Thus * Z[c1; c2; : :]: L()* and L()* is generated by the coefficients of the formal group law whose logarithm series is t2 tk t + c1__ + . .+.ck-1 __+ . .:. 2 k But this is precisely Lazard's universal formal group law; see, for exam- ple, [14]. We review the method of Milnor [21] for constructing polynomial generators for the Lazard ring L* = L()*. This enables us to throw some light on the extension * L*. If we let ( p; if n + 1 is a power of the prime p, hn = 1; if n + 1 is a not a prime power, n+1 then hn is the highest common factor of the integers i , for 1 i n. We may therefore choose integers nisuch that Xn n + 1 ni = hn: i=1 i We then let n X un = niOE(i; n - i + 1) 2 L2n: i=1 Lazard's theorem asserts that L* = Z[u1; u2; : :]:. For example one choice of the nileads to u1 = OE1; u2 = 1_2(OE2 - OE21); u3 = _1_12(OE3 + 2OE1OE2 - 3OE31): 18 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY By (6.4)we can write 1 OE(i; n - i + 1) = ____________ OEn + zi(OE) i!(n - i + 1)! where zi(OE) is an integer polynomial in the OEr, for r < n. Writing n for the integer (n + 1)!=hn, we have Xn 1 n + 1 nun = ni___ OEn + zi(OE) i=1 hn i which shows that nun 2 2n and that (6.9) nun OEn mod decomposables in *. Thus if we let fn = nun, the fn are alternative polynomial generators for *, with the inclusion * = Z[f1; f2; : :]: L* = Z[u1; u2; : :]: given by fn 7! nun. For a general delta operator E the universal morphism : ! E induces L() : L() ! L(E). If we write L()*(un) = vn 2 E2n, then L(E)* is generated over E* by the elements Xn vn = nie(i; n - i + 1): i=1 In general there are relations among the vn. Consider the delta operator =OE1 = Z[OE2; OE3; : :]:; D + OE2D3=3! + . . .. Here (6.10) L(=OE1)* = Z[u1; u2; u3; : :]:=(u1) = Z[u2; u3; : :]:; since OE1 = u1. Other the other hand, (6.11) L(=OE2)* = Z[u1; u2; u3; : :]:=(2u2 + u21); since OE2 = 2u2 + u21in L*. Delta operators arising from topology are always Leibniz, for the map j :S3 ! CP 1 which we discussed in x2 is, up to homotopy, a map of H-spaces, so in these cases (E)* is already a Hopf algebra. Since the universal delta operator is not Leibniz it does not stem from a spectrum. Yet its Leibniz extension L* = L()* is isomorphic to the com- plex bordism ring MU* [23], and so corresponds to the case E = MU, the universal complex-oriented spectrum. It is this relationship which allows us to investigate Leibniz extensions by methods adapted from algebraic topol- ogy. In general the extension E* L(E)* may be very complicated. We conclude this section by giving a specific example, and by proving a result which shows that there are general divisibility relations which must hold among the en in L(E)*. EXTENSIONS OF UMBRAL CALCULUS II 19 Example 6.12. We refer to the quadratic delta operator R = Z[u]; D + uD2=2 as the Bessel operator, since the associated sequence is made up of graded Bessel polynomials; see [30]. This delta operator is not Leibniz; for example, since r1 = u, with ri = 0 for i 2, by (4.2), br1(x) = x br2(x) = 1_2(x2 - ux); br3(x) = 1_6(x3 - 3ux2 + 3u2x); so that br1(x)br2(x) = 3br3(x) + 2ubr2(x) + 1_2u2br1(x): Thus u2=2 2 L(R)*. The associated formal group law is p ________ p ________ X + Y + u-1 1 + 2uX - 1 1 + 2uY - 1 : Hence ui+j-1 r(i; j) = (-1)i+jCi-1Cj-1_______; 2i+j-2 where Cn is the Catalan number 1 2n Cn = ______ : n + 1 n By analysing the 2-divisibility of such numbers it is not hard to show that L(R)* is multiplicatively generated by the elements uj=2j-1, where j is of the form 2m + 2n - 1. The following Kummer congruence for the coefficients of a delta operator is related to Theorem 2 of [8]. It is only the first of a series of such result* *s, but it is sufficient for the applications in x9. We intend to return to the general case elsewhere. Theorem 6.13. If p is prime, then en+p-1 enep-1 mod p in the ring L(E)*. Proof. We will in fact show that the congruences hold modulo p in the subring Z[c1; c2; : :]:of L(E)*which is generated by the elements cn defined by (6.8). Lagrange inversion applied to the equations (2.1)and (6.8)yields X (n + )! en = (-1) _____________________________ck11ck22. .c.kss; 2k13k2. .(.s + 1)ksk1!k2! . .k.s! where the sum is over all sequences (k1; k2; : :;:ks) such that k1 + 2k2 + . .+.sks = n, and = k1 + k2 + . .+.ks. Writing e(ck11ck22. .c.kss) for the coefficient of ck11ck22. .c.kssin en, we can rearrange this formula to give s n + Y (t + 1)kt ! e(ck11ck22. .c.kss) = (-1) ___________: 2k1; 3k2; : :;:(s + 1)kst=1(t + 1)ktkt! We will show that (6.14) e(ck11ck22. .c.ksscp-1) e(ck11ck22. .c.kss) mod p in Z[c1; c2; : :]:. This will be sufficient, since we will also show that (6.15) ep-1 cp-1 mod p: The following result is trivial; see x1 of [9] for the proof of stronger res* *ults. 20 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Lemma 6.16. If p is prime and m; a 1, then 8 ><1 mod p; if m = 1, (ma)!_ (-1)a mod p; if m = p, maa! >: __ 0 mod p; if m - p and ma > p. |__| It follows that e(ck11ck22. .c.kss) 0 mod p unless (t + 1)kt < p for all t 6= p - 1. Hence the congruence (6.14)holds for all monomials for which (t + 1)kt > p for some t 6= p, in particular for all monomials divisible by ct for some t p. Consider now the multinomial coefficient n + : 2k1; 3k2; : :;:pkp-1 If each term (t + 1)kt is less than p for t < p - 1, then this multinomial coefficient will be divisible by p if n + - pkp-1 p. Thus the con- gruence (6.14) holds (again because both sides are zero modulo p) for all monomials ck11ck22. .c.kp-1p-1for which 2k1 + 3k2 + . .+.(p - 1)kp-2 p: Together with Wilson's theorem, this argument also shows that congru- ence (6.15)holds. For the remaining cases, Lemma 6.16 shows that e(ck11ck22. .c.kp-1+1p-1) (n + + p)(n + + p - 1) . .(.n + + 1) e(ck11ck22. .c.kp-1p-1)_______________________________________ (pkp-1 + p)(pkp-1 + p - 1) . .(.pkp-1 + 1) modulo p. Now we know that pkp-1 < n + < pkp-1 + p, so that we may cancel the factor pkp-1 + p in the fraction, leaving a numerator and denominator which are, by Wilson's theorem, both congruent to -1. This shows that e(ck11ck22. .c.kp-1+1p-1) e(ck11ck22. .c.kp-1p-1) mod p __ and completes the proof. |__| Corollary 6.17. For any delta operator E, j-1 epj-1 e1+p+...+pp-1 mod p __ in L(E)*. |__| 7. Leibniz extensions and double delta operators In this section we discuss how the Leibniz properties of E.F are influenced by the corresponding properties of its constituent components E and F . Given a double delta operator G, equation (6.2)provides us with two sets of elements g1(i; j) = g1(i; j; 1) and g2(i; j) = g2(i; j; 1) in GQ* correspond- ing to the two delta operators 1G and 2G. Definition 7.1. A double delta operator G is Leibniz if both 1 and 2 are Leibniz delta operators EXTENSIONS OF UMBRAL CALCULUS II 21 Thus G is Leibniz if and only if all the g1(i; j) and the g2(i; j) belong to G*. If G fails to be Leibniz, we define the Leibniz extension G* L(G)* GQ* by adjoining the elements g1(i; j) and g2(i; j). Lemma 7.2. Given a double delta operator G, the delta operator 1G is Leibniz if and only if 2G is Leibniz. Proof. As we remarked in x3, the two penumbral coalgebras (1G)* and (2G)* are the equal. Hence if one is closed under multiplication so is the_ other. |__| Corollary 7.3. If G fails to be Leibniz, we may form L(G)* by adjoining __ only one of the two sets of elements g1(i; j) and g2(i; j). |__| Corollary 7.4. Given two delta operators E and F , L(E) . F = L(E . F ) = E . L(F ): In particular, if E is Leibniz, then E . F = E . L(F ), and if F is Leibniz, then E . F = L(E) . F ; in both cases E . F is Leibniz. Proof. It suffices to note that, if G = E . F , then we have g1(i; j) = e(i;_j) and g2(i; j) = f(i; j). |__| The concept of a Leibniz double delta operator is precisely equivalent to a pair of formal group laws over a torsion-free ring together with a strict isomorphism between them. As we remarked in x6, the double delta operators arising from complex- oriented ring spectra, in the way described at the end of x3, are always Leibniz. We discuss now their relationship with the `dot' construction of x4. Recall from x2 that if E and F are complex-oriented ring spectra with torsion-free coefficient rings E* and F*, they each give rise to a delta oper- ator, also denoted by E and F . We write (E . F )* for the domain of the double delta operator E . F . We can think of this ring is an algebraic model for the ring (E ^ F )* ~=E*(F ) ~=F*(E). Under certain conditions the two are isomorphic. We consider first the universal case where E = F = MU. The ring MU*(MU) is torsion-free, so there is a double delta operator de- fined over it, which has the structure of a (1MU ; MU1 )-operator; see the discussion at the end of x3. Theorem 7.5. The morphism (MU . MU)* ! MU*(MU) provided by Proposition 4.9 is an isomorphism of rings. Proof. By Definition 4.4, (MU . MU)* is generated over MU* MU* by the coefficients of the series expressing 1MU in terms of MU1 . But this is precisely the description of MU*(MU) given, for example, in [2, Part II], where these coefficients, which we will denote by the traditional b1; b2;_: :,: are shown to be polynomial generators over MU*. |__| 22 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Corollary 7.6. If the ring ( . )* is identified with the polynomial algebra *[b1; b2; : :]:as in Proposition 4.8, and MU*(MU) is identified with the algebra MU*[b1; b2; : :]:, as in the proof of Theorem 7.5, then the morphism ( . )* ! MU*(MU) provided by Proposition 4.10 is the homomorphism __ induced by the inclusion * L()* = MU*. |__| Proposition 7.7. For complex-oriented spectra E and F such that the co- efficient rings E* and F* are torsion-free, (E . F )* ~=E* MU* MU*(MU) MU F*: Proof. By Proposition 4.11, (E . F )* ~=E* * ( . )* * F*: Now Corollary 7.6 shows that E* * ( . )* ~=E* MU* MU*(MU), and __ by symmetry a similar result holds on the right. |__| The complex orientations of E and F give rise to maps of spectra MU ! E and MU ! F , and hence to a map MU ^ MU ! E ^ F . This gives an MU*-bialgebra homomorphism MU*(MU) ! E*(F ) and hence, by Propo- sition 4.11, there is an algebra homomorphism E;F :(E . F )* ! E*(F ): If E*(F ) is torsion-free, then there is a (e; f)-operator E*(F ); l; r . In this case E;F is the homomorphism * given by Proposition 4.9. Recall that the complex-oriented spectrum G is said to be Landweber exact if the homology theory G*( ) can be defined for all spaces X as G*(X) = G* MU* MU*(X): Criteria on G* for this to hold were set down by Landweber in [16]; we discuss them at the beginning of x10. Examples of such spectra include complex K-theory, the elliptic spectrum Ell, the Johnson-Wilson spectra E(n), the Brown-Peterson spectra BP and the complex bordism spectrum MU itself. Proposition 7.8. If one of E or F is Landweber exact, then E;F : (E . F )* ! E*(F ) is an isomorphism. In all cases E;F is a monomorphism and a rational isomorphism. Proof. If F is any complex-oriented ring spectrum, then (see [2, Part II]) MU*(MU) MU* F* ~=MU*(F ): Hence Proposition 7.7 shows that (E.F )* is isomorphic to E*MU* MU*(F ). If E is Landweber exact, then we have an isomorphism (E . F )* ~=E*(F ). If F is Landweber exact, we can argue analogously. For any spectrum F , the rationalisation F Q is always Landweber exact, so that (E . F )* Q = (E . F Q)* ~=E*(F Q) = E*(F ) Q; showing that E;F is a rational isomorphism, and hence a monomorphism, __ since (E . F )* is torsion-free. |__| EXTENSIONS OF UMBRAL CALCULUS II 23 Note that Corollary 4.6 gives an example where E;F fails to be an iso- morphism. It appears that, as in this case, E;F frequently induces an isomorphism between (E . F )* and E*(F )=torsion. Since (E . F )* is torsion- free, by Proposition 7.8, this is so whenever E;F maps onto E*(F )=torsion. In particular, this happens in the examples E = H and F = BP J, where J denotes any subset of a set of polynomial generators of BP*, and BP J is the BP -module spectrum with homotopy BP*=(J); see, for example, the calculations of [15]. We shall need the following result in x10. Proposition 7.9. If E is Landweber exact, then E*(F ) is a flat F*-module. Proof. The argument is essentially the same as that of [19, Remark 3.7]. By Propositions 7.7 and 7.8, E*(F ) ~=E* MU* MU*(MU) MU F*: Thus the functor E*(F ) F* _ can be written as the composition of the functors MU*(MU) MU* _ and E* MU* _. The first of these is exact because MU*(MU) is a flat MU*-module, and is a functor into the category of MU*(MU)-comodules. But the second is exact on this category; this is __ ensured by the Landweber exactness conditions [16]. |__| 8. Stably penumbral polynomials and K-theory The Leibniz delta operator K arises in algebraic topology from complex K-theory, and in combinatorics and classical numerical analysis as the dis- crete derivative (for example see [24]). It is defined by K* = Z[u; u-1] and k = u-1(euD - 1), where u has dimension 2. In particular, the coefficients of k are given by kn = un. It has Bkn(x) = x(x - u) . .x.- (n - 1)u as its associated sequence, and therefore the normalised version may be writ- ten as bkn(x) = un x=un. Thus K is Leibniz by virtue of the Vandermonde convolution identity x=u x=u X l j x=u = ; i j l j l - i l see, for example, [29]. Recall, for a torsion-free delta operator F , the penumbral coalgebra (F )* of x2. As an F*-module it is free on generators b0; b1; b2; : :,:where we identify b0 with the unit element 1 and b1 with x, the generator of the original Hopf algebra F*[x]. If F is Leibniz, so that (F )* is in particular closed under multiplication by x, we may construct the localised F*-algebra (F )*[x-1] as the limit of the directed system of modules (F )* x-!(F )* x-!(F )* x-!. .;. where the maps x are multiplication by x. Theorem 8.1. If F is a Leibniz delta operator, with F* torsion-free, then (F )*[x-1] and (K . F )* are isomorphic as F*-algebras. Proof. By Definition 4.4, the F*-algebra (K . F )* is generated as an algebra over K* F* = F*[u; u-1] by the umbral elements bfn(k). The nature of the coefficients kn = un means that bfn(k), as defined in (4.1), is just the 24 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY polynomial u-1bfn(u). Hence (K . F )* is multiplicatively generated over F*[u; u-1] by the polynomials bfn(u). But, since F is Leibniz, (F )* is closed under multiplication, thus any product of the polynomials bfn(x) can be written as an F*-linear combination of the bfn(x). Hence (K . F )* is additively generated over F*[u; u-1] by the polynomials bfn(u). Now define the F*-module homomorphism ff: (F )* ! (K.F )* by setting ff bfn(x) = bfn(u). Clearly ff is a monomorphism of rings, with ff(x) = u. Hence the diagram (F )* --x-! (F )* ? ? ff?y ?yff (K . F )*--u-! (K . F )* commutes. Since multiplication by u is an isomorphism on (K . F )*, we obtain a map of F*-algebras (F )*[x-1] ! (K . F )*; __ which is clearly an isomorphism. |__| In the case of a general torsion-free delta operator which is not necessarily Leibniz, we need to consider the minimal Leibniz extension. Corollary 8.2. If E is a torsion-free delta operator, then L(E) *[x-1] and (K . E)* are isomorphic as L(E)*-algebras. Proof. By Corollary 7.4, (K . E)* = K . L(E) *. It is this identification which gives (K . E)* the structure of an L(E)*-algebra. The result is then_ the special case of Theorem 8.1 in which F = L(E). |__| 9.Hattori-Stong theorems The classical Hattori-Stong theorem [13, 32], for which we will give a proof as Theorem 9.10, tells us, in Hattori's formulation, that the MU*-module map MU* ! K*(MU) induced by the right unit MU ! K ^ MU is the inclusion of a direct summand [2, Part II, x14]. Note that MU* is a direct summand as a subgroup of the abelian group K*(MU). As we show in Proposition 10.6, it is not a summand as an MU*-module. What Hattori actually shows is that if ff 2 MU* is divisible by an in- teger m in K*(MU), then ff is already divisible by m in MU*. In other words MU* is a pure subgroup of K*(MU); see [11, Ch. IV]. The connec- tive K-theory group k*(MU) = (k . MU)* lies between MU* and K*(MU). Since k*(MU) is finitely generated in each degree, it follows that the pure subgroup MU* is a summand of k*(MU), and, since k*(MU) is a summand of K*(MU), so is MU*. Such finiteness arguments may not be available for a general delta operator, so we will phrase our generalisations in terms of the concept of purity. We can interpret MU* as L* = L()* and K*(MU) as (K . MU)* = K . L() * = (K . )*. For any torsion-free delta operator E, Corollary 8.2 shows that E* L(E)* (K . E)* so the Hattori-Stong theorem motivates EXTENSIONS OF UMBRAL CALCULUS II 25 us to ask when L(E)* is a pure subgroup of (K . E)*. In order to discuss this question we consider the smallest pure subgroup containing L(E)*. Definition 9.1. Let (E)* = EQ* \ (K . E)* denote the rational closure of E* in (K . E)*. Thus (E)* consists of all ff 2 (K . E)* for which there exists a non-zero integer m such that mff 2 E*. Since E* L(E)* EQ*, it is equivalent to ask that there is an integer m such that mff 2 L(E)*. It is also clear that (E)* is a subring of (K . E)*, and that L(E)* is a subring of (E)*. The Hattori-Stong theorem leads us to ask under what conditions (E)* is equal to L(E)*. When (E)* = L(E)*, so that L(E)* is a pure subgroup of (K . E)*, we shall refer to this result as the Hattori-Stong theorem for the delta operator E. If G is a double delta operator, we may identify the rings (K . 1G)* and (K . 2G)*. Since L(1G)* = L(2G)* (see Lemma 7.3), the Hattori-Stong theorem holds for 1G if and only if it holds for 2G. We will study the ring (E)* by using Corollary 8.2 to identify (K . E)* with the ring L(E) *[x-1]. If we write Xn (9.2) xn = oe(n; r)ber(x); r=1 in L(E) *, the coefficients oe(n; r) 2 L(E)* may be computed as e r fi nff oe(n; r) = ( ) fix : In the notation of [26], oe(n; r) = r!SE (n; r), where SE (n; r) is an E-theory Stirling number of the second kind. The leading coefficient oe(n; n) is equal to n!, while oe(n; 1) = en-1 , the coefficient of Dn=n! in e; see (2.1). In fact Proposition 3.3, applied to the double delta operator H . E, gives X r ie1 jm1 ie2j m2 ek mk oe(n; r) = n! __ __ . . .________ ; m1; m2; : :;:mk 2! 3! (k + 1)! where the summation is over all sequences of finite length (m1; m2; : :;:mk) such that m1 + 2m2 + . .+.kmk = n - r, and m1 + m2 + . .+.mk r. If m1+m2+. .+.mk = r-s, then the coefficient of the monomial em11em22. .e.mkk in oe(n; r) is equal to (9.3) n(n - 1) . .(.n - s + 1) n - s r : 2;_:_:;:2-z__"; 3;_:_:;:3-z__"; : :;:k_+_1;_:_:;:k-+z1_____"m1; m2* *; : :;:mk; s m1 m2 mk This formula shows that oe(n; r) 2 E*, as we expect since xn 2 (E)* L(E) *. The penumbral coalgebra L(E) * is a free L(E)*-module on the poly- nomials ber(x), so that the divisibility of an element l of L(E)* in (K . E)* = L(E) *[x-1], and hence in (E)*, depends on the divisibility properties of the loe(n; r) in L(E)*. The next result gives a criterion for divisibility by a prime. 26 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Proposition 9.4. Let p be a prime and l 2 L(E)*, then p divides l in (E)* if and only if p divides lenp-1in L(E)* for some non-negative integer n. Proof. Definition 9.1 shows that p | l in (E)* if and only if p | l in (K.E)* = L(E) *[x-1]. This in turn holds if and only if p | lxm in L(E) * for some non-negative integer m, which, using the expansion (9.2), is equivalent to p | loe(m; r) for 1 r m. j-1 Suppose that p | lenp-1in L(E)*, then p | le1+p+...+pp-1for some suffi- ciently large j. Now Corollary 6.17 shows that p | lepj-1 = loe(pj; 1). We will show that p divides all the coefficients oe(pj; r) for r > 1. It is clear that if s > 0 in (9.3), then oe(pj; r) is divisible by p. While if s = 0, the first multinomial coefficient is pj 2;_:_:;:2-z__"; 3;_:_:;:3-z__"; : :;:k_+_1;_:_:;:k-+z1_____" m1 m2 mk which is divisible by p in every case except that corresponding to oe(pj; 1). Conversely, if p - lenp-1for all n, then by the same reasoning as above __ p - lepj-1 = loe(pj; 1) for all j. Hence p - l in (E)*. |__| The Hattori-Stong theorem (that L(E)* is equal to (E)*) amounts to saying that if a prime p divides l 2 L(E)* in (E)*, then p already divides l in L(E)*. Hence Proposition 9.4 gives us a criterion for when the Hattori- Stong theorem applies. Theorem 9.5. The Hattori-Stong theorem holds for the delta operator E if and only if, for all primes p and all l 2 L(E)*, whenever p divides lep-1 in L(E)*, then p divides l in L(E)*. Proof. If (E)* 6= L(E)*, then there must be a prime p and an element l 2 L(E)* such that p divides l in (E)* but p does not divide l in L(E)*. By Proposition 9.4, p divides lenp-1for some n. Applying the condition of the statement n times shows that p must divide l in L(E)*, which is a contradiction. Conversely, if (E)* = L(E)* and p divides lep-1 in L(E)*, then Propo- __ sition 9.4 with n = 1 shows that p divides l in L(E)*. |__| Applying Proposition 9.4 in the case l = 1 will tell us which primes are invertible in (E)*. Theorem 9.6. The prime p is invertible in (E)* if and only if p di- __ vides enp-1in L(E)* for some non-negative integer n. |__| Theorem 9.6 raises the question of which primes are invertible in L(E)*. Proposition 6.7 shows that L(E)* is multiplicatively generated over E* by elements of positive degree. It follows that if En = 0 for n < 0, then no new relations can be introduced in degree 0, so that p is invertible in L(E)* if and only if it is invertible in E*. On the other hand, if we invert the two-dimensional generator of the Bessel delta operator R to give R[u-1] = Z[u; u-1]; D + uD2=2 we have L R[u-1] * = Z[1=2][u; u-1] since u2=2 2 EXTENSIONS OF UMBRAL CALCULUS II 27 L(R)4; see Example 6.12. Hence 2 is invertible in L R[u-1] *, but not in R[u-1]*. If the prime divisibility structure of the ring L(E)* is reasonably simple, we can make some simplification of Theorems 9.5 and 9.6. Definition 9.7. A ring R has unique integer factorisation if, for all r; s 2 R and prime integers p, whenever p divides rs, then either p divides r or p divides s. Proposition 9.8. Assume that L(E)* has unique integer factorisation, then the Hattori-Stong theorem for E holds if and only if, for all primes p, either p is invertible in L(E)* or p does not divide ep-1 in L(E)*. Proof. In the presence of unique integer factorisation p | lep-1 if and only if p | l or p | ep-1. Now the criterion of Theorem 9.5 says that if p - ep-1,_then p divides any element of L(E)*. |__| It is useful to phrase what is essentially the same result in a different wa* *y. Proposition 9.9. If L(E)* has unique integer factorisation, then (E)* is the localisation of L(E)* in which those primes p which divide ep-1 are_ inverted. |__| We conclude this section by considering a number of examples. Firstly we can give a simple proof of Hattori and Stong's original result. Theorem 9.10 (The classical Hattori-Stong theorem). ()* = L()*. Proof. Since L()* is a polynomial ring over Z, it has unique integer factori- sation and no primes are invertible in L()*. Hence, by Proposition 9.8, we need only show that p - OEp-1 in L()*. There are many ways of doing this. We could remark that the congruence (6.9)shows that OEp-1 is congruent modulo decomposables to (p - 1)!up-1, where up-1 is one of the polynomial generators of L()*. Alternatively, the morphism of delta operators ! K given by the universality of maps OEn to un 2 K*. Since up-1 is indivisible_ in K* = L(K)*, the prime p cannot divide OEp-1 in L()*. |__| We may abstract the second of these arguments as follows. Proposition 9.11. Suppose that the Hattori-Stong theorem holds for the delta operator F , and we are given a morphism E ! F of delta operators. If L(E)* has unique integer factorisation, and the same primes are invertible in each of the rings L(E)* and L(F )*, then the Hattori-Stong theorem holds for E. Proof. We apply Proposition 9.8. Supposing that the prime p divides ep-1 in L(E)*, we deduce that p divides fp-1 in L(F )*, so that the Hattori-Stong__ theorem for F implies that p is invertible in L(F )* and hence in L(E)*. |__| Proposition 9.12. For the Artin-Hasse delta operator A of Example 3.7, (A)* = A* Z(p) 28 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Proof. We apply Proposition 9.9. The delta operator A is Leibniz; see [14, x3.2]. Note that the series (3.8) giving D in terms of a is at the same time a divided power series with coefficients in A* and a power series with coefficients in A*[1=p]. It follows that the same is true for the inverse series for a in terms of D. Hence aq-1 is divisible by q!=pp(q!). So for a prime q 6= p, the coefficient aq-1 is divisible by q. But ap-1 = -(p - 1)!v which_is not divisible by p. |__| We consider now two examples of delta operators where E* L(E)* (E)* EQ* are all proper inclusions. Recall from (6.10)that for the delta operator =OE1 we have L(=OE1)* = Z[u2; u3; : :]:, which has unique integer factorisation. Since OE1 = 0 in L(=OE1)*, the prime 2 is invertible in (=OE1)*. On the other hand, for an odd prime p, we saw in (6.9)that (p - 1)!up-1 OEp-1 modulo decom- posables in L*, so that p does not divide OEp-1 in L(=OE1)*, hence p is not invertible in (=OE1)*. Thus by Proposition 9.9 (=OE1)* = Z[1_2][u2; u3; : :]:: Our second example is the delta operator =OE2 for which L(=OE2)* = Z[u1; u2; u3; : :]:=(2u2 + u21) (see (6.11)) does not have unique integer fac- torisation. However unique integer factorisation fails only for the prime 2, and since OE21= -2u2 in L(=OE2)* Theorem 9.6 shows that 2 is invertible in (=OE2)*. Theorem 9.6 also shows that 3 is invertible, since OE2 = 0 in L(=OE2)*. For all other primes we may use the argument which applied to the previous example to conclude that (=OE2)* = Z[1_6][u1; u3; : :]:: 10. Topological examples We conclude by examining topological Hattori-Stong theorems. We sup- pose that E is a complex-oriented spectrum with E* torsion-free. The corre- sponding delta operator E will already be Leibniz, and, since the K-theory spectrum is Landweber exact, Proposition 7.8 shows that (K . E)* is iso- morphic to K*(E). Thus the Hattori-Stong theorem for E-theory asserts that E* is isomorphic to its rational closure (E)* under the right unit E* ! K*(E). We recall the conditions for a spectrum to be Landweber exactn[16]. Fix- ing a prime p, for n 0 let un 2 E2pn-2 be the coefficient of tp in the p-series [p]E (t) of the E-theory formal group law. Clearly u0 = p. The ex- actness conditions at the prime p are that p; u1; u2; : :i:s a regular sequence in the ring E*. That is, for all n 0, multiplication by un on the quo- tient E*=(p; u1; : :;:un-1 ) should be injective. This is required to hold for all primes p. The first of these conditions says that multiplication by p is injective on E*, and thus E* is torsion-free, which is a blanket assumption for all the spectra we consider. The next condition, at height one, says that multiplication by u1 on E*=(p) is injective. That is to say, if p divides u1e in E*, then p divides e. EXTENSIONS OF UMBRAL CALCULUS II 29 Now u1 cp-1 modulo p, where cp-1 is the coefficient of tp=p in the log series of the formal group law (see [17, Lemma 2.1]), and we saw in (6.15) that cp-1 ep-1 mod p in E*. So, given that L(E)* = E*, this height-one condition is equivalent to the criterion of Theorem 9.5 for the Hattori-Stong theorem to hold. We shall say that E* (or more generally an E*-module M) satisfies the height-one Landweber exactness condition for all primes if E* (or M) is torsion-free and for each prime p the sequence p; ep-1 is regular. We have thus proved Theorem 10.1. If E is a complex-oriented ring spectrum with E* torsion- free, then E* is a pure subgroup of K*(E) if and only if E satisfies the __ height-one Landweber exactness condition for all primes. |__| Corollary 10.2. If E is a complex-oriented ring spectrum which is Landwe-_ ber exact, then E* is a pure subgroup of K*(E). |__| The following generalisation closely parallels a result of Laures [18, The- orem 1.2.3] which applies to the case of elliptic cohomology. Theorem 10.3. Let E be a complex-oriented ring spectrum and X a space or spectrum such that E*(X) satisfies the height-one Landweber exactness condition for all primes, then E*(X) is a pure subgroup of K*(E ^ X). Proof. Since E*(CP 1) = (E)* is a free E*-module, there is a K"unneth isomorphism E*(X ^ CP 1) ~= E*(X) E* (E)*. Similarly, since Propo- sition 7.9 shows that K*(E) = E*(K) is a flat E*-module, K*(E ^ X) = E*(X ^ K) ~=E*(X) E* K*(E). Moreover the isomorphism of Theorem 8.1 is compatible with these isomorphisms so that K*(E ^ X) ~= E*(X) E* (E)*[x-1]. We may now apply the arguments used in the proofs of Propo- __ sition 9.4 and Theorem 9.5. |__| It is striking that for these results we only need the first two of Landwe- ber's criteria. It is tempting to suspect that using the higher conditions one might prove that E* is a pure subgroup of F*(E), where both E and F are Landweber exact theories. In the absence of any analogue of Theorem 8.1, or indeed of any space to play the role that CP 1 plays for K-theory, it is difficult to see how to generalise our proofs. Of course, if E*(X) satisfies the height-one exactness condition, then so does E*; the converse is true if E*(X) is free over E*. This follows in turn if X is a finite complex and H*(X) is a free Z-module. Smith [31] states the classical Hattori-Stong theorem, for E = MU, in this last form. Suppose that E satisfies the height-one exactness condition, and M is a flat E*-module, then tensoring the exact sequences ep-1 0 ! E* p-!E* and 0 ! E*=(p) ---! E*=(p) with M, we find that M satisfies the height-one exactness condition. Hence Proposition 7.9 shows that E*(E) is a pure subgroup of K*(E ^ E) if E is Landweber exact. More generally a similar result will hold for E*(E ^ E ^ . .^.E). These remarks follow closely the case of elliptic cohomology considered in Theorem 2.3.1 of [18]. 30 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY Though Theorem 10.1 gives the complete picture, in some cases the fol- lowing results provide a simple way to verify that the Hattori-Stong theorem holds. Proposition 10.4. Suppose that the complex orientation MU ! E extends via a map E ! K to an orientation of K, and E* has unique integer factorisation, then the Hattori-Stong theorem holds for E-theory. Proof. The K-theory orientation which factors through E may not be the standard orientation. But together the two orientations determine a double delta operator over K*. For the standard orientation we consider for each prime p the sequence p; up-1 which is certainly regular in K* = Z[u; u-1], so the Hattori-Stong theorem holds. But, as we remarked in x9, this implies that the Hattori-Stong theorem holds for the delta operator determined by the other orientation. Since no primes are invertible in K*, none can be__ in E*, so Proposition 9.11 applies. |__| For p-local spectra there is a corresponding result. We let G denote a ring spectrum which is a summand of p-local K-theory. Proposition 10.5. Suppose that E is a complex-oriented ring spectrum for which E* is a Z(p)-module. If the complex orientation MU ! E extends via a map E ! G to an orientation of G, and E* has unique integer factorisa- tion, then the Hattori-Stong theorem holds for E-theory. Proof. We remark that a suitable orientation for G gives rise to the delta operator A Z(p), where A is the Artin-Hasse operator of Example 3.7. Proposition 9.12 shows that the Hattori-Stong theorem holds for G-theory. __ The remainder of the proof follows that of Proposition 10.4. |__| It is clearly possible to state results which are intermediate between Propositions 10.4 and 10.5, for example for theories in which 2 is invert- ible and which map into KO[1_2]. We conclude by proving the result referred to at the beginning of x9. Proposition 10.6. Any splitting map K*(MU) ! MU* will not be an MU*-module map. Proof. Suppose otherwise, then the identity on MU* factors through MU*- module maps MU* ! K*(MU) ! MU*: Now apply _MU* H* to this sequence and we obtain a factorisation of the identity H* ! K*(MU) MU* H* = K*(H) ! H*: However, H* = Z, concentrated in degree 0 while K*(H) is a rational vector_ space as shown by Propositions 4.7 and 7.8. |__| We are unaware of any such splitting having been explicitly written down. EXTENSIONS OF UMBRAL CALCULUS II 31 References [1]J. F. Adams, On Chern characters and the structure of the unitary group, Pr* *oc. Cambridge Philos. Soc., 57 (1961), 189-199. [2]J. F. Adams, Stable homotopy and generalised homology, University of Chicag* *o Press, Chicago, 1974. [3]J. F. Adams, A. S. Harris and R. M. Switzer, Hopf algebras of cooperations * *for real and complex K-theory, Proc. London Math. Soc. (3) 23 (1971), 385-408. [4]A. Baker, Combinatorial and arithmetic identities based on formal group law* *s, Al- gebraic topology, Barcelona, 1986, 17-34, Lecture Notes in Math., 1298, Spr* *inger, Berlin-New York, 1987. [5]A. Baker, F. Clarke, N. Ray and L. Schwartz, The Kummer congruences and the stable homotopy of BU, Trans. Amer. Math. Soc. 316 (1989), 385-432. [6]J. Blissard, Theory of generic equations, Quart. J. Pure Appl. Math. 4 (186* *1), 279- 305. [7]J. M. Boardman, The eightfold way to BP -operations or E*E and all that, Cu* *rrent trends in algebraic topology, Part 1 (London, Ont., 1981), 187-226, CMS Con* *f. Proc., 2, Amer. Math. Soc., Providence, RI, 1982. [8]L. Carlitz, Some properties of Hurwitz series, Duke Math. J. 16 (1949), 285* *-295. [9]F. Clarke, The universal von Staudt theorems, Trans. Amer. Math. Soc. 315 (* *1989), 591-603. [10]A. Frohlich, Formal groups, Lecture Notes in Math., 74, Springer, Berlin-He* *idelberg, 1968. [11]L. Fuchs, Abelian groups, Hungarian Acad. Sci., Budapest, 1958. [12]A. Hasse, Die Gruppe der pn-prim"aren Zahlen f"ur einen Primteiler p von p,* * J. Reine Angew. Math. 176 (1936), 174-183. [13]A. Hattori, Integral characteristic numbers for weakly almost complex manif* *olds, Topology 5 (1966), 259-280. [14]M. Hazewinkel, Formal groups and applications, Academic Press, New York, 19* *78. [15]Fam Ngok An'Kyong, On orders of k-invariants in cobordism theories with sin* *gular- ities, Soviet Math. Dokl. 31 (1985), 362-364. [16]P. S. Landweber, Homological properties of comodules over MU*(MU) and BP*(B* *P ), Amer. J. Math. 98 (1976), 591-610. [17]P. S. Landweber, Supersingular elliptic curves and congruences for Legendre* * poly- nomials, Elliptic curves and modular forms in algebraic topology, Princeton* *, 1986, 69-83, Lecture Notes in Math., 1326, Springer, Berlin-New York, 1988. [18]G. Laures, The topological q-expansion principle, MIT Thesis, 1996. [19]H. R. Miller and D. C. Ravenel Morava stabilizer algebras and the localisat* *ion of Novikov's E2-term, Duke Math. J. 44 (1977), 433-447. [20]H. Miller, Universal Bernoulli numbers and the S1-transfer, Current trends * *in alge- braic topology, Part 2 (London, Ont., 1981), 437-449, CMS Conf. Proc., 2, A* *mer. Math. Soc., Providence, RI, 1982. [21]J. Milnor, On the cobordism ring * and a complex analogue (part I), Amer. J* *. Math. 82 (1960), 505-521. [22]W. Nichols and M. Sweedler, Hopf algebras and combinatorics, Umbral calculu* *s and Hopf algebras (Norman, OK, 1978), 49-84, Contemp. Math., 6, Amer. Math. Soc* *., Providence, RI, 1982. [23]D. Quillen, On the formal group laws of unoriented and complex cobordism th* *eory, Bull. Amer. Math. Soc. 75 (1969), 1293-1298. [24]N. Ray, Extensions of umbral calculus: penumbral coalgebras and generalised* * Bernoulli numbers, Adv. Math. 61 (1986), 49-100. [25]N. Ray, Symbolic calculus: a 19th century approach to MU and BP , Homotopy * *theory (Durham, 1985), 195-238, London Math. Soc. Lecture Note Ser., 117, Cambridge Univ. Press, Cambridge-New York, 1987. [26]N. Ray, Stirling and Bernoulli numbers for complex oriented homology theori* *es, Alge- braic topology (Arcata, CA, 1986), 362-373, Lecture Notes in Math., 1370, S* *pringer, Berlin-New York, 1989. 32 FRANCIS CLARKE, JOHN HUNTON, AND NIGEL RAY [27]N. Ray, Loops on the 3-sphere and umbral calculus, Algebraic topology (Evan* *ston, IL, 1988), 297-302, Contemp. Math., 96, Amer. Math. Soc., Providence, RI,19* *89. [28]N. Ray, Universal constructions in umbral calculus, to appear in the Procee* *dings of the Rotafest, April 1996. [29]J. Riordan, Combinatorial identities, Krieger, Huntington, NY, 1979. [30]S. Roman, The umbral calculus, Academic Press, Orlando, 1984. [31]L. Smith, A note on the Stong-Hattori theorem, Illinois J. Math. 17 (1973),* * 285-289. [32]R. E. Stong, Relations among characteristic numbers I, Topology 4 (1965), 2* *67-281. [33]M. E. Sweedler, Hopf algebras, W. A. Benjamin, New York, 1969. Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales E-mail address: F.Clarke@Swansea.ac.uk Department of Mathematics and Computer Science, University Road, Le- icester LE1 7RH, England E-mail address: jrh7@mcs.le.ac.uk Department of Mathematics, Manchester University, Manchester M13 9PL, England E-mail address: nige@mathematics.manchester.ac.uk