aa r X i v : . [ m a t h . N T ] D ec POLYNOMIALS OF BINOMIAL TYPE AND LUCAS’ THEOREM
DAVID GOSS
Abstract.
We present various constructions of sequences of polynomials satisfying the Bi-nomial Theorem in finite characteristic based on the theory of additive polynomials. Variousactions on these constructions are also presented. It is an open question whether we thenhave accounted for all sequences in finite characteristic which satisfy the Binomial Theorem. Introduction
Inspired by classical work on p -adic measure theory, and, in particular, the connectionwith the Mahler expansion of continuous functions, we discussed measure theory in finitecharacteristics in [Go1]. In characteristic p , an analog of the Mahler expansion was given byWagner [Wa1] using the basic Carlitz polynomials (see Definition 6 below). By construction,the Carlitz polynomials, which are created out of additive functions via digit expansions,satisfy the Binomial Theorem as a consequence of Lucas’ famous congruence (see Theorem2). This then allowed the author, together with Greg Anderson, to compute the associatedconvolution measure algebra which is then isomorphic to the algebra of formal divided powerseries (Definition 2 below), as opposed to the ring of formal power series classically given byMahler’s result.Inspired further by the present work of Nguyen Ngoc Dong Quan [DQ1], we have recentlyrevisited our results from the viewpoint of the umbral calculus and the theory of sequencesof polynomials satisfying the Binomial Theorem. It is indeed quite remarkable that theclassical generating function of such a sequence (see Subsection 2.2) is also an element of thealgebra of divided power series.In this note we show how the Carlitz construction allows us to obtain a very large sub-space of sequences satisfying the Binomial Theorem. We also show that this space is closedunder multiplication in the algebra of divided power series. Moreover, still using additivepolynomials, we are able to construct many other examples of sequences satisfying the Bino-mial Theorem which do not arise from the Carlitz method; indeed, this second constructionappears to be largely complimentary to that of Carlitz. We discuss various actions on thespace of divided elements and how they relate to our constructions. Along the way, we derivea decomposition of the divided power series associated to Dirac measures (see Remarks 3below). It is now an open and very interesting question whether we then have constructedenough sequences of binomial type out of additive functions to generate all of them.Finally, in Remarks 6, we present an umbral construction of the Carlitz module due to F.Pellarin.We thank Nguyen Ngoc Dong Quan, Federico Pellarin, and Rudy Perkins for their helpin the preparation of this note. Date : December 10, 2014. General Theory
Basic notions.
Let F be a field and let F [ x ] be the polynomial ring in one indetermi-nate over F . Let P := { p i ( x ) } ∞ i =0 be a sequence of polynomials. Definition 1.
We say that P satisfies the Binomial Theorem if and only if p n ( x + y ) = P ni =0 (cid:0) ni (cid:1) p i ( x ) p n − i ( y ), for all n ≥
0. We let B be the set of all sequences satisfying theBinomial Theorem.Obviously { x i } ∞ i =0 ∈ B as does the trivial sequence P := { } ∞ i =0 . Another standard,nontrivial, example is the sequence { ( x ) n } where ( x ) n is the Pochhammer symbol defined as( x ) n := x ( x − x − · · · ( x − n + 1). For more, we refer the reader to [RT1].I thank Nguyen Ngoc Dong Quan for the proof of the following simple result. Proposition 1.
Let P = { p i ( x ) } ∞ i =0 ∈ B be nontrivial. Then p ( x ) ≡ .Proof. Assume first that p ( x ) is not identically 0. Note that, by definition, p ( x + y ) = p ( x ) p ( y ). Upon setting y = 0 we obtain p ( x ) = p ( x ) p (0). By assumption, p ( x ) isnonzero for some x ; thus p (0) = 1. On the other hand, p ( x − x ) = p ( x ) p ( − x ) = p (0) = 1;as p ( x ) is a polynomial, we conclude conclude that it is constant. Thus p ( x ) is identically1. It remains to conclude that p ( x ) is not identically 0. But if it does identically vanish, asimple use of the Binomial Theorem and induction, establishes that all p i ( x ) are also trivialcontradicting our assumption. (cid:3) We let B ∗ ⊂ B be the subset of nontrivial sequences.2.2. The generating function.
Let { D i } ∞ i =0 be the (nontrivial) divided power symbolswith the property that D j · D j := (cid:18) i + jj (cid:19) D i + j . (1)Thus in characteristic 0 one may view D i as x i /i ! and in all characteristics one can view D i as the divided differential operator given by D i x n := (cid:18) ni (cid:19) x n − i . (2) Definition 2.
Let R be a commutative ring with unit. We let R be the commutativering of formal series P a i D i with the obvious addition and multiplication.We topologize R by using the descending chain of ideals M i = { D i , D i +1 . · · · } forwhich it is complete.Now let H = { h i ( x ) } be an arbitrary sequence of polynomials in F [ x ]. Definition 3.
We set f H ( x ) := ∞ X i =0 h i ( x ) D i ∈ F [ x ] . (3)The following result is well-known and the proof follows immediately from the definitions. OLYNOMIALS OF BINOMIAL TYPE AND LUCAS’ THEOREM 3
Proposition 2.
We have f H ( x + y ) = f H ( x ) f H ( y ) (4) if and only if H ∈ B . Let BD ∗ be the set { f P ( x ) } P ∈ B ∗ ; in this case, by Proposition 1, we know that f P ( x ) =1+ { higher terms in D } . Theorem 1.
The set BD ∗ forms an abelian group under multiplication.Proof. Let W = { w i ( x ) } where w ( x ) = 1 and the rest vanish. Clearly W ∈ B ∗ and f W ( x ) =1 ∈ F [ x ] . Now let P, H be two members of B ∗ and let g ( x ) := f P ( x ) f H ( x ); noticethat clearly the coefficients of g are polynomials in x . Note further that the commutativityof F [ x ] immediately implies that g ( x + y ) = g ( x ) g ( y ), which shows that BD ∗ is closedunder multiplication.It remains to show that every element f P ( x ) in BD ∗ is invertible. As f ( x ) ≡
1; we canwrite f P ( x ) = 1+ ˆ f P ( x ). Expanding 1 / (1+ ˆ f P ( x )) by the geometric series, which converges inthe topology of F , gives an element with polynomial coefficients inverse to f P ( x ) (cid:3) Remarks . The classical theory of polynomials of binomial type has applications in fieldsas diverse as combinatorics and even quantum field theory (see, for instance, [Ki1]).3.
The Theory in finite Characteristic and Lucas’ Theorem
For the rest of this paper we assume that F has characteristic p > Lucas’ Theorem.
Let q = p λ where λ ≥
1. Let m = P m i q i and n = P n j q j be twointegers written q -adically. Suppose m ≥ n . Theorem 2. ( Lucas ) We have (cid:18) mn (cid:19) ≡ Y i (cid:18) m i n i (cid:19) (mod p ) . (5) Proof.
By definition we have ( x + y ) m = m X e =0 (cid:18) me (cid:19) x e y m − e . (6)But in characteristic p we also have( x + y ) m = Y i ( x q i + y q i ) m i . (7)The result follows upon completing the multiplication, equating 6 and 7, and noting theuniqueness of the q -adic expansions. (cid:3) First Results in Finite Characteristic.
Let f ( x ) ∈ F [ x ]. Recall that f ( x ) is additive if and only if f ( x + y ) = f ( x ) + f ( y ) for all x and y in the algebraic closure of F . It is simpleto see that this happens if and only if f ( x ) only contains monomials of the form cx p j .Our first result gives a basic connection between elements P ∈ B and additive polynomials. Proposition 3.
Let P = { p i ( x ) } ∈ B . Then p p j ( x ) is additive for all j ≥ .Proof. In the Binomial Theorem all the lower terms are congruent to 0 mod p so that theresult follows immediately from the definition of P . (cid:3) DAVID GOSS
Proposition 4.
Let g ( x ) = 1 + { higher terms } ∈ F [ x ] . Then g ( x ) p ≡ .Proof. This is an immediate consequence of Lucas’ Theorem 2. (cid:3)
Corollary 1.
The abelian group BD ∗ is naturally an F p -vector space.Remarks . Let f P ( x ) ∈ BD ∗ . By Proposition 2 we have f P ( px ) = f P (0) = 1 = f P ( x ) p , giving another proof of the proposition in this case.3.3. The Carlitz Construction.
Let q = p λ as before and assume that F q ⊆ F . Carlitzturned Lucas’ Theorem around to construct certain sequences (Definition 6 below) satisfyingthe Binomial Theorem which we generalize in this subsection.Let E := { e j ( x ) } ∞ j =0 be a sequence of F q -linear polynomials (so each e j ( x ) is a finite linearcombination of monomials of the form x q n ). Let L q be the set of such sequences; note that L p is obviously an F -vector space. Let i be a nonnegative integer written q -adically as P mt =0 i t q t . Definition 4.
Let E ∈ L q as above. We set p E,i ( x ) := Y t e t ( x ) i t . (8)Let P E be the sequence { p E,i ( x ) } . The next proposition is then basic for us. Proposition 5.
The sequence P E ∈ BD ∗ .Proof. We need to compute p E,i ( x + y ) = Q t e t ( x + y ) i t = Q t ( e t ( x ) + e t ( y )) i t . The resultnow precisely follows upon multiplying out and using Lucas’ Theorem 2. (cid:3) Proposition 5 is our first construction in characteristic p of polynomials satisfying theBinomial Theorem. Example . For i ≥ ℓ q ( i ) be the sum of its q -adic digits. Then the sequence { x ℓ q ( i ) } ofelements in F [ x ] satisfies the Binomial Theorem. Indeed, this is just Proposition 5 where all e t ( x ) = x .3.4. The Connection with Nonarchimedean Measure Theory.
Let A := F q [ θ ] and,for t ≥ A ( t ) := { a ∈ A | deg( a ) < t } ; notice that A ( t ) is an F q -vector spaceof dimension t . In this subsection F will be a field containing A . Definition 5.
Let t ≥ D t be the product of all monic elements ofdegree t and e t ( x ) := Y a ∈ A ( t ) ( x − a ) . (9)It is easy to see that e t ( x ) is additive (indeed, F q -linear). We set E := { e t ( x ) /D t } ∞ t =0 , andnow let i = P mt =0 i t q t be written q -adically as above. Definition 6.
We define the
Carlitz polynomials P E = { p E,i ( x ) } by p E,i ( x ) := Y t (cid:18) e t ( x ) D t (cid:19) i t , where e t ( x ) is given in Equation 9. OLYNOMIALS OF BINOMIAL TYPE AND LUCAS’ THEOREM 5
It is traditional to set G := P E and G i ( x ) := p E,i ( x ). Carlitz has shown that G i ( a ) ∈ A for all a ∈ A and by Proposition 5 we know that G satisfies the Binomial Theorem. For anontrivial prime v of A we let A v be the completion of A at v . C. Wagner [Wa1] has shownthat G forms an orthonormal Banach basis for the space of continuous functions from A v toitself, see also [Co1]. Using the fact that G satisfies the Binomial Theorem, the author, andGreg Anderson, noticed that this implies that the convolution algebra of Nonarchimedeanmeasures on A v with values in A v is then isomorphic to A v ; see [Go1]. Example . Let α ∈ A v . The Dirac measure at α , δ α , is defined by R A v f ( x ) dδ α ( x ) = f ( α )for all continuous f ( x ) (note that in non-Archimedean analysis this construction gives atrue, bounded, measure). We know that f ( x ) can be expressed as P c i G i ( x ) for a uniquesequence { c i } of scalars which tends to 0 as i → ∞ . Thus, we have δ α corresponds to P i G i ( α ) D i ∈ A v . Moreover, by definition, the convolution of δ α and δ β is δ ( α + β ) ,and this precisely corresponds to Proposition 2 in the description of measures as elements of A v .3.5. The Carlitz Construction is closed under Multiplication of Generating Func-tions.
We now return to having F be an arbitrary field with F q ⊂ F . By Corollary 1, thegroup BD ∗ is an F p -vector space. Definition 7.
We let BD ∗ q, ⊆ BD ∗ be the subset of those generating functions that arisefrom L q by the Carlitz construction Proposition 5.Let W = { w t } and V = { v t } be two elements of L p with sum W + V . Let P W , P V and P ( W + V ) be the corresponding sequences of polynomials satisfying the Binomial Theorem with f P W ( x ), f P V ( x ) and f P ( W + V ) ( x ) the corresponding generating functions. Theorem 3. In F [ x ] we have f P W ( x ) · f P V ( x ) = f P ( W + V ) ( x ) . Proof.
Let i = P t i t q t be written q -adically as before. By definition p ( W + V ) ,i ( x ) = Y t ( w t ( x ) + v t ( x )) i t . One now expands out and uses Lucas as before. The result is exactly the element obtainedby multiplying f P W ( x ) and f P V ( x ) and the result follows. (cid:3) Corollary 2. BD ∗ q, ⊆ BD ∗ is an F p -subspace.Proof. The space BD ∗ q, is the image of L p under the linear injection E P W . (cid:3) Corollary 3.
Let E ∈ L q and let − E be its inverse obtained by negating all its elements.Then f P E ( x ) · f P − E ( x ) = 1 . Example . Returning to the case of Example 2, the measure theoretic version of Corollary3 is precisely the standard fact that the convolution of δ α and δ − α is δ = 1. Remarks . Theorem 3 implies that all elements of BD ∗ q, may be expressed as a, possiblyinfinite, convergent product over those elements created out of only one nonzero additivefunction. In particular, we derive such a product decomposition for the Dirac measures δ α aselements of A v . In fact, when α is not an element of A , this product decomposition has DAVID GOSS infinitely many terms. In turn, this product decomposition corresponds measure theoreticallyto expressing δ α as an “infinite convolution,” a concept which is well-known in classicalmeasure theory, see for example [EST].4. A Second Construction
A Counterexample.
Corollary 2 states that BD ∗ q, is a subspace of BD ∗ . In the nextexample we will produce an example of an element of BD ∗ not lying in BD ∗ q, for any q . Example . Let { f i ( x ) } ∞ i =1 be a collection of non-trivial additive functions. Let f ( x ) =1 + P ∞ i =0 f i ( x ) D p i − . Then f ( x ) ∈ BD ∗ \ BD ∗ q, . Indeed, f ( x + y ) = f ( x ) f ( y ) by theadditivity of the f i ( x ) and the fact that for i, j > D p i − · D p j − = 0 (due to the vanishingof the binomial coefficient); thus f ( x ) ∈ BD ∗ by Proposition 2. As the coefficients of D p t , t >
0, vanish, it can not be in BD ∗ q, for any q . Remarks . It is easy to see that the element f ( x ) of the above example can be expressedas the infinite product Q i (1 + f i ( x ) D p i − ). Note also that multiplying f by an element of BD ∗ p, also can not lie in BD ∗ p, as it is a group.4.2. A Second Construction of Elements in BD ∗ . Example 4 leads to a very generalconstruction of elements of BD ∗ which is, in some sense, complimentary to the Carlitzconstruction of Subsection 3.3. Definition 8.
We say a sequence X := { D i j } ∞ j =0 of divided elements is a null sequence ifand only if D i j · D i t = 0 for all j and t (where j = t is permitted).Now let E := { e j ( x ) } ∞ j =0 ∈ L p be a sequence of additive functions and define f X,E ( x ) = 1 + X j e j ( x ) D i j . Proposition 6.
The element f X,E ( x ) ∈ BD ∗ .Proof. This follows exactly as in Example 4. (cid:3)
Note, as before, we readily deduce the convergent product f X ,E ( x ) = Y j (1 + e j ( x ) D i j ) . (10)Let π X : X → BD ∗ be defined by π X ( E ) := f X,E ( x ). The next result then follows imme-diately. Proposition 7.
The mapping π X is an injection of L p into BD ∗ . The Action of the Group S ( q ) . In [Go2] we introduced the group S ( q ) of homeomor-phisms of Z p to itself. The construction of the group is very simple; let σ be a permutationof the set { , , . . . } and let y ∈ Z p be written q -adically as P i =0 y i q i . One then defines σ ∗ y = P i y i q σi . Note that this action preserves the positive integers. Furthermore, weestablished that, in characteristic p >
0, the induced mapping σ ∗ ( D j ) := D σ ∗ j is an auto-morphism of R . Proposition 8.
The automorphism σ ∗ stabilizes both BD ∗ and BD ∗ q, . OLYNOMIALS OF BINOMIAL TYPE AND LUCAS’ THEOREM 7
Proof.
Since σ ∗ is a ring homomorphism, the first statement follows from Proposition 2. Thesecond statement follows because σ ∗ merely changes the order of the additive functions inthe Carlitz construction which is inessential. (cid:3) Corollary 4.
Let f ( x ) be the function of Example 4 and let f σ be its image under σ ∗ . Then f σ ∈ BD ∗ \ BD ∗ q, .Remarks . Let X = { D i j } be as in Subsection 4.2 and let X σ = { D σ ∗ i j } . Then clearly σ ∗ ◦ π X = π X σ .4.4. Further Actions.
Federico Pellarin has kindly pointed out three other actions on thealgebra R in finite characteristic. Definition 9.
Let f = P a i D i .1. We set π ( f ) := P a pi D I .2. We set π ( f ) := P a i D pi .3. Let r ∈ R . We set π ( f ) := P a i r i D i (the “evaluation” map).It is easy to see that all three of these maps are endomorphisms of R with the firstbeing an obvious automorphism in the case where R is a perfect field. They therefore take BD ∗ to itself. It is also straightforward to see that these endomorphisms stabilize BD ∗ p, and that the q -th power of the first two actions stabilizes BD ∗ q, . Their induced actions onthe maps π X of Subsection 4.2 are also easy to compute.4.5. Final Remarks.
We have seen that both the Carlitz construction (Corollary 2) andour second construction in Subsection 4.2 give rise to injections of the spaces L q into BD ∗ . Question 1.
Do the images of the above injections generate BD ∗ ?In other words, do the additive polynomials ultimately account for all elements of B ? Notethat in characteristic 0, the elements of B are described by the Bell Polynomials [RT1].
Remarks . The umbral theory of [RT1], with its “black magic” of linear maps etc., hasfurther connections with the arithmetic of function fields as pointed out by F. Pellarin andwhich we briefly describe here. Let C be the Carlitz module and let ω ( t ) be the Anderson-Thakur function as in [Pe1]. Let τ be the q -th power mapping acting as in [Pe1] whereone defines (Definition 2.6) the polynomials b j ( t ) by τ j ω ( t ) = b j ( t ) ω ( t ). It is readily seenthat b j ( t ) = Q j − e =0 ( t − θ q e ) (and in fact, as shown in ibid, these polynomials are universal inthat the coefficients of both the Carlitz exponential and logarithm may be easily expressedusing them). Now define the A -linear map from A [ τ ] → A [ t ] by τ i b i ( t ). This gives anisomorphism with the inverse given by t j C θ J as one deduces from the theory of ω ( t ). Inparticular, we derive another construction of the Carlitz module and another indication ofits ubiquity. References [Co1]
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