The p-adic Z-transform

Let a+pnZp be a ball in Zp and assume that a is the smallest natural number contained in the ball. We define a measure z on Zp by putting z(a + pnZp) = za 1-zpn where z E Cp, Iz 1. Let f be a continuous function defined on Zp. The mapping f -; is similar to the classical Z-transform. We use this transform to give new proofs of several known results : the Mahler expansion with remainder for a continuous function, the Van der Put expansion, the expansion of a function in a series of Sheffer polynomials. We also prove some new results. 1991 Mathematics subject classification : 46S10


Introduction
Let Zp be the ring of p-adic integers, where p is a prime. Qp and Cp denote, as usual, the field of the p-adic numbers and the completion of the algebraic closure of Qp. ~.~ denotes the normalized p-adic valuation on Cp. We start by defining a measure on Zp . Let a + p"Zp be a ball in Zp. We may assume that a is the smallest natural number contained in the ball. Our measure will depend on a parameter z E Cp. . It is well-known that this defines a distribution on Zp. Let D denote the set {z E Cp | |z -1| > 1}.
An easy calculation shows that if z E D then 1. Throughout this paper we will assume that z E D. Hence z is a measure. Now let f : Zp -; Cp be a continuous function.
If we associate with f the integral F(z) we get a transformation that we call the p-adic Z-transform since it is similar to the classical Z-transform used by engineers.
The aim of this paper is to show how this transform can be used to obtain a number of results in p-adic analysis. In section 2 we start by studying the integral F(z). In sections 3 and 4 we use the p-adic Z-transform to give new proofs of several known results : the Mahler expansion with remainder for a continuous function, the Van der Put expansion, the expansion of a function in a series of Sheffer polynomials. In section 5 we use the results of section 2 to find approximations to the p-adic logarithm of 2. We prove e.g. that the following congruence is valid in Zp This integral has already been studied and used by Y. Amice and others in [1] and [4].
A fundamental property of this integral is Proposition : F( z ) is an analytic element in D (in the sense of Krasner). This means that F(z) is the uniform limit of a sequence of rational functions with poles outside D. But, by definition F(z) = Zp f ( x )z ( x ) = l i m 0 3 A 3 p n -1 k = 0 f ( k ) z k 1 -z p n (1) It is not diflicult to show that the sequence in (1) is uniformly convergent. Since the zeroes of 1 -zpn are outside D, F(z ) is an analytic element in D. Corollary : F satisfies the "principle of analytic continuation" i.e. if F(z) is zero on a ball in D it is zero in the whole of D.
The fact that .F(z) is an analytic element in D is very useful in proving properties of the integral (1). As an example we prove that = f(0) + z + in D (2) Proof : For |z| 1 formula (1) reduces to The trivial identity can be written as This is a priori valid for |z| 1. By analytic continuation it is valid in D.
We now list some properties of the integral f We only give a few indications about the proofs. Proof : Replace f(x) by /(.r -1) in (2) to get Iteration of this formula yields (5).
which is obvious. The formula is valid in D by analytic continuation.
where denotes the sup-norm.
Remark : It follows from (5)  Formula (6) shows that F(z) belongs to B if f E C(Zp).
Hence it makes sense to consider the mapping We will call F(z) the p-adic z-transform of f for the following reason. If Izl 1 then F(z) = In applied mathematics it is customary to call the "generating func-k=0 tion" F(z) the z-transform of f.
We now examine the properties of the z-transform.
It is easily verified that T is linear and continuous.
This proves that T is injective.
T is also surjective. To see this we start from a given F(z) = with lim ak = 0. It follows from (6) that the z-transform of the function f(x) = f a k { x k } is equal to the given F(z) since (0394kf)(0) = ak.
Although we do not need it in the sequel we will also prove that T is an isometry. For this we need a lemma.
Hence a thus and the lemma is proved.
Proposition : T is an isometry.  This was obtained in [5] by a different method.
Remark : Until now we have assumed that the functions of C(Zp) take their values in Cp. If we replace Cp by a field that is complete for a non archimedean valuation containing Qp, the method still works. The only restriction is that we can no longer use any property whose proof uses analytic continuation.

The expansion of a continuous function in a series of Sheffer polynomials
In this section we will use the p-adic z-transform to generalize the main theorem of [6]. We first recall a few elements of the p-adic umbral calculus developed in [6].
Let R be a linear continuous operator on C(Zp, K), where K is a field containing ~F that is complete for a non archimedean valution. If R commutes with E it can be written in 00 the form R == ~~ where is a bounded sequence in.K. The result that we want to ==o generalize is the following.
Proposition [6]  Formula (6) shows that the measure obtained in this way is the measure introduced in section 1.

00
Now let Q = y~ and S = ~ be two operators commuting with E where S is i=O t==0 invertible.
If bo = 0, any operator R, commuting with E, can be written in the form 00 R = 03A3 rnQn, rn ~ K n=0 We can see this as an equality between operators or as an identity between formal power series in A. If we take R = the coefficients rn will depend on z. Let us write it in the form Writing out everything as a powerseries in A and comparing the coefficient of 0394n we see that Tn(z ) is a polynomial of degree n in z. If, moreover, |b1| === 1 the sequence is bounded.
Multiplying (16) with 5~ and applying the operators on both sides to a function f 6 C(Zp,R') we get the series This series is uniformly convergent since lim = 0. n~T he idea is now to take the inverse z-transform of (17). Now the z-transform of (x n) is zn (1-z)n+1. Hence the z-transform of a polynomial of degree n is of the form where is also a polynomial of degree n. Taking the inverse transform of (17) we get where is a polynomial of degree n. This is the expansion we wanted to obtain.
To see that (18) is a generalization of (15) take S equal to the identity operator and take / equal to the polynomial pn in (15). (18) then reduces to pn(.r) = tn(x). In the general case the polynomials are called "Sheffer polynomials" in umbral calculus.
In this section we show that it is possible to refine this result using the properties of the integral studied in section 2.
The idea is to construct approximations for the integral on the LHS of (20). This will yield the following theorem. where ên = (-1)nW" i c -21-1 1 g 2' P (-1)k+1 k -8~n 2 ( (-1)k+1 2k+1 For the proof we need the value of a few integrals. We collect these results in the following lemma. i denotes a squareroot of -1. (1) These are special cases of formula (9).
(2) These are special cases of (8) with z = i.
The sum is extended over all primitive q-th roots of unity 8 with ~ ~ 1. g is an integer prime to p.
In [1] the author supposes that g is a prime but this restriction is not necessary. Clearly the LHS of (21) Is independant of g. Taking respectively q = 2 and q = 4 we get If k remains in a fixed residue class mod (p -1) the LHS of (21) is a continuous function of k. Hence (21) and (22) remain valid for negative integers (except possibly for k = -1). Taking k = -3 we get Since (8)  Neglecting the last term we see that (b) is proved. To obtain {c) it is sufficient to take a linear combination of (24) and (25) such that the integral ( ~-1{x) disappears.