Factorial-Type Recurrence Relations and p -adic Incomplete Gamma Functions

We introduce an automorphism S of the space C ( Z p , C p ) of continuous functions Z p → C p and show that it can be used to give an alternative construction of the p -adic incomplete Γ-functions recently introduced by O’Desky and Richman [7]. We then describe various properties of the automorphism S , showing that it is self-adjoint with respect to a certain non-degenerate symmetric bilinear form deﬁned in terms of p -adic integration, and showing that its inverse plays a role in a p -adic integral-transform space akin to the role of diﬀerentiation in the classical space of Laplace-transformed functions. Some consequences for p -adic incomplete Γ-functions are given along the way.


Introduction
In number theory, one is interested in finding p-adic analogues of classical functions of a real or complex variable.Important examples are p-adic L-functions and Morita's p-adic Γ-function.These p-adic analogues are usually defined via some interpolation property.More precisely, if f is a complex function whose values at the positive integers can be viewed also as elements of the field C p , via a choice of isomorphism C ∼ = C p , say, then one looks for a continuous p-adic function F : Z p → C p that takes the same values as f on Z ≥1 .By the denseness of Z ≥1 in Z p , the continuous function F , if it exists, is uniquely determined.In practice, the function f may need to be modified first, and the set of points at which one interpolates may be some other subset of Z dense in Z p .
In a recent paper [7], O'Desky and Richman construct a p-adic analogue of the incomplete Γ-function Γ(−, r) for each r ∈ 1 + pZ.Their construction uses the combinatorial machinery of derangements.
Here, we introduce an automorphism S of the space C(Z p , C p ) of continuous functions, restricting to an automorphism of the space la(Z p , C p ) of locally analytic functions Z p → C p , and show that O'Desky and Richman's p-adic incomplete Γ-function Γ p (−, r) can be constructed by applying S −1 to the locally analytic p-adic function x → r x (r ∈ 1 + pZ).We thus give a construction of O'Desky and Richman's functions that does not use derangements.
We then turn to proving properties of the automorphism S. We show in Section 4 that it is self-adjoint with respect to a non-degenerate symmetric bilinear form on C(Z p , C p ) defined in terms of p-adic integration.We also show that, with respect to a convolution product ⋆ on C(Z p , C p ), S again enjoys a sort of self-adjointness; see Section 5.
Next, in Section 6, we introduce an integral transform I : C(Z p , C p ) → la(Z p , C p ), again related to p-adic integration, describe its relationship with S, derive an integral-transform formula for the p-adic incomplete Γ-functions, and give a complete description of the image of I .We also prove a convolution property of I .
Finally, in Section 7, we relate S to solutions of the well-known differential equation F ′ + F = G.For example, if G ∈ Z[[t]], then the unique solution F ∈ Z p [[t]] can be constructed concretely via S −1 .We work in a more general setting than Z p .The precise result is Proposition 7.3.
Running throughout is the function q = S −1 (1), where 1 : x → 1.Up to a certain constant multiplicative factor and a one-unit shift in the argument, q is the p-adic analogue of Γ(−, 1).It features prominently in results on the convolution product ⋆ and in the description of the image of I .The function q has been well studied in various contexts (see Section 2.4 for a sample of works), and we hope our contribution will provide further evidence that it may have a role to play in p-adic analysis.

Notation and definitions
Throughout, | • | and v p will denote, respectively, the absolute value and the valuation on C p , normalized such that |p| = 1/p and v p (p) = 1.We will occasionally make reference to the balls B <ρ (a, K) = {x ∈ K | |x − a| < ρ}, where K is a subfield of C p , a ∈ K, and ρ is a positive real number.
We will also refer often to the constant function 1 : Z p → C p , x → 1.
Let C(Z p , C p ) denote the C p -vector space of continuous functions Z p → C p .We define a map S : C(Z p , C p ) → C(Z p , C p ) as follows.If φ ∈ C(Z p , C p ), then S(φ) is the function Z p → C p defined by It is clear that S(φ) is again continuous.
Proof.The continuity of φ follows from the assumption of uniform convergence.For the second assertion, observe that for x ∈ Z p , Note that φ is a well-defined function in C(Z p , C p ), because its sequence of Mahler coefficients converges to 0. Then by Proposition 2.2, Theorem 2.4 The map S is an automorphism of the C p -vector space C(Z p , C p ).
Proof.For injectivity, suppose that S(φ) = 0, i.e., φ(x) = xφ(x − 1) for all x ∈ Z p .We show by induction on n that φ(n) = 0 for all n ∈ Z ≥0 , from which it follows that φ = 0 by continuity.The base case is clear, because φ(0) = 0 • φ(−1) = 0. Now let n ≥ 0 and assume that φ(n) = 0. Then φ(n + 1) = (n + 1)φ(n) = 0.This completes the induction and therefore the proof of injectivity.For surjectivity, let ψ : Z p → C p be any continuous function, and let ψ = ∞ k=0 b k β k be its Mahler series, so that b k → 0. We already know from Lemma 2.3 that Because the functions φ k are all maps into Z p , they have sup-norm φ k ≤ 1, so the sequence of functions b k φ k converges uniformly to zero.Hence, by uniform convergence, we have a well-defined, continuous function As for the formula in (2.1), if x ∈ Z p , then the interchanging of the summation signs being justified as follows.Fix x ∈ Z p and let it is enough to show that, for any ǫ > 0, there are only finitely many pairs
The following equivalent form of Proposition 2.5 is an immediate consequence of the fact that, if (x n ) n is a sequence of real numbers, then the sequence (inf k≥n x k ) n is monotone non-decreasing.
We use Proposition 2.6 to prove the following.
Theorem 2.7 The map S restricts to an automorphism of the C p -vector space la(Z p , C p ).
Proof.It is clear from the definition of S that S(φ) ∈ la(Z p , C p ) if φ is.It is also clear from the injectivity of S as a map on C(Z p , C p ) that it is still injective when restricted to la(Z p , C p ).It remains to show that if ψ : Z p → C p is locally analytic, then the unique φ ∈ C(Z p , C p ) such that S(φ) = ψ is also locally analytic.
Let the Mahler coefficients of ψ be (b n ) n≥0 .By Theorem 2.4, the nth Mahler coefficient of φ is The assumption that ψ is locally analytic says, by Proposition 2.6, that there are β > − 1 p−1 and N ∈ Z ≥0 such that, for all n ≥ N , Assume now that n ≥ N , and write where Let us first consider the case where β ≥ 0, in which case , which tends to 0 as n → ∞.Therefore, if α is any negative real number, v p (a ′ n ) ≥ nα for n sufficiently large.Thus, φ is locally analytic by Proposition 2.6, because we may choose α such that −1/(p − 1) < α < 0.
We may thus assume for the remainder of the proof that β < 0. Then because v p (s)/n → 0, there is Further, each b ′ k in the sum in (2.3) has valuation greater than or equal to kβ because k ≥ N in the sum.But kβ ≥ nβ, because β < 0 and k ≤ n for k in the sum.Thus, the summation term in (2.3) has valuation at least nβ as well.Therefore, v p (a ′ n ) ≥ nβ for all n ≥ max(N, K), and so φ is locally analytic by Proposition 2.6.

An alternative description of
and the real numbers |n!|M converge to 0 as n → ∞.Therefore, by uniform convergence, we have a well-defined, continuous function Thus, S(φ) = ψ, so S −1 (ψ) = φ.

Relationship to factorial-type recurrence relations
Consider a first-order linear recurrence relation where h, ψ : Z p → C p are continuous functions and the a n are in C p .Let us say that a solution (a n ) n≥0 to such a recurrence relation is continuous if there is a continuous function φ : Z p → C p such that φ(n) = a n for all n ≥ 0. By an automatically seeded recurrence relation, we will mean a recurrence relation as in (2.4) for which the function h has a zero n 0 ∈ Z.Note that n 0 may be positive, negative, or zero.The terminology is explained via the following proposition and its proof.Proposition 2.9 If an automatically seeded recurrence relation admits a continuous solution, then the solution is unique.
In particular, the value a n0 in an automatically seeded recurrence relation admitting a continuous solution is determined by the recurrence relation itself, unlike in an ordinary recurrence relation, where a n0 may typically be chosen freely.
For us, the function h of interest will be h(x) = x, so that the recurrence relation becomes a n = na n−1 +ψ(n).It seems reasonable to call this a factorialtype recurrence relation, because one recovers the usual factorial recurrence relation by taking ψ = 0. Theorem 2.4 says that every factorial-type recurrence relation a n = na n−1 + ψ(n), where ψ ∈ C(Z p , C p ), has the unique continuous solution a n = S −1 (ψ)(n).Thus, every time we invoke the inverse map S −1 in this paper, we are essentially considering the unique continuous solution to some factorial-type recurrence relation.Although this point of view will not be emphasized, the reader may well like to keep it in mind while reading the rest of the paper.

The function q
Central to this paper is the function q = S −1 (1), where 1 : . By definition, it satisfies q(x) = xq(x − 1) + 1 for all x ∈ Z p and therefore gives the unique continuous solution to the recurrence relation a n = na n−1 + 1.We note as well that it is locally analytic by Theorem 2.7.(See also [8], where the function is called A.) This function has been studied many times already, in several contexts and with various symbols to denote it.Aside from the reference [8] just mentioned, we point out just a few more: [5, Lemma 5], where a connection to the incomplete Γ-function Γ(−, 1) is made, and [10] and [2], which consider convergents to e, the latter proving a conjecture of Sondow.

p-adic incomplete Γ-functions
In [7], O'Desky and Richman define a p-adic incomplete Γ-function Γ p (−, r) for each r ∈ 1 + pZ p via the combinatorial machinery of derangements.Their function satisfies a p-adic interpolation property with respect to the usual (complex) incomplete Γ-function Γ(−, r).We give a construction using instead the invertible map S : locally analytic by Proposition 2.5.Let γ r = S −1 (g r ), also locally analytic by Theorem 2.7.Define p to be p if p is odd and to be 4 if p = 2, and define again obviously locally analytic, because γ r is.The function γ r will give us the desired interpolation property.By definition of S and g r , i.e., exp(r p)γ r (x) = x exp(r p)γ r (x − 1) + r x exp(r p) for all x ∈ Z p , i.e., γ r (x + 1) = xγ r (x) + r x exp(r p) for all x ∈ Z p .
In particular, Consider, then, the unique field map τ p : Q(e) → Q p mapping e −1 to exp(p).
(The map τ p appears already in O'Desky and Richman's paper; we do not claim τ p as our own.)Applying τ p to both sides of (2.6), we have Thus, the values τ p (Γ(n, r)), n ∈ Z ≥1 , satisfy the same recurrence relation as the values γ r (n), as in (2.5).Further, Remark.We point out a small discrepancy between our function γ r and O'Desky and Richman's Γ p (−, r) in the case where p = 2.They define the map τ p : Q(e) → Q p by sending e −1 to exp(p), while our choice is to map e −1 to exp(p).The maps are the same when p is odd, but our choice is made to ensure that the exponential series exp(p) converges even in the case p = 2, where p = 4. (Note that exp(x) converges if and only if |x| < |p| 1/(p−1) .)The exceptional nature of the prime 2 often requires the introduction of an extra factor of 2, especially in Iwasawa theory.See [11,Sect. 7.2], for example.Another option might be to employ a continuous extension of exp to C 2 , so that exp( 2) is defined for that extension.However, there is no canonical such extension.See [9, Chap.5, Sect.4.4] for a discussion.
We will provide an integral-transform formula for γ r in Section 6.2.1.

Recovering O'Desky and Richman's formula
Since γ r and Γ p (−, r) interpolate the same p-adic values at the positive integers (with the above remark about the p = 2 case understood), they must be the same function, and indeed, the following formula for γ concords with that given for Γ p (−, r) in [7, Theorem 1.1].

1-Lipschitz functions
Recall that a function φ : Proof.First, suppose that φ : Z p → A is 1-Lipschitz.It is clear from the definition of S(φ) that its image is again in A. For the 1-Lipschitz property of S(φ), we take x, y ∈ Z p .Then by the assumption that the image of φ is contained in A.
Conversely, suppose that ψ : Z p → A is 1-Lipschitz, and let φ = S −1 (ψ), so that φ(x) = xφ(x − 1) + ψ(x) for all x ∈ Z p .It is clear from Proposition 2.8 that φ has its image in A. As for the 1-Lipschitz property of φ, it is equivalent to the statement that for all a, x ∈ We first show by induction on x that if x ∈ Z ≥0 , then φ(x + a) ≡ a φ(x).For the case x = 0, observe that We now know that φ(x + a) ≡ a φ(x) for all x ∈ Z ≥0 , and it remains to extend this to x ∈ Z p , which we may achieve by continuity.Specifically, the function maps Z ≥0 into the set V = {α ∈ A | |α| ≤ |a|} and therefore, being continuous, maps Z p into V as well because V is closed in A.
We apply the above to the functions γ r with r ∈ 1 + pZ p .
Proof.Recall that γ r = S −1 (g r ) where g r (x) = r x .By Proposition 2.12, the statement to be proven is equivalent to the statement that g r is 1-Lipschitz.For that, note that if x, y ∈ Z p , and if n ≥ 0, then This is a weak form of the lemma in [9, Chap.5, Sect.1.5].Hence, We claim that max n∈Z ≥1 n|r − 1| n ≤ 1.Indeed, this inequality holds if and only if n|r 3 Generating functions the ordinary and exponential generating functions, respectively, of (φ(n)) n≥0 .We also let The second equality is equivalent to Proof.Note that the equality and For example, let us apply the proposition to the functions γ r = S −1 (g r ) appearing in Section 2.5.
Proof.The equalities follow immediately from the proposition and the straightforward observations that G * ,gr (t) = exp(rt), G gr (t) = 1 r − t .

Self-adjointness of S
Recall that a linear functional µ : In the following, we use the term measure to mean a bounded linear functional µ : Our approach is similar to that of [6, Chap.4, Sec.1], except that measures there are assumed to have values in {x ∈ C p | |x| ≤ 1}.We do not impose this restriction.
If µ is a measure, then the power series ) Conversely, a power series G(t) = ∞ n=0 b n t n with bounded coefficients defines a measure µ satisfying where φ ∈ C(Z p , C p ) has Mahler coefficients (a n ) n≥0 .The measure µ associated to G in this way will be denoted µ(G).
In the special case where φ : x → (1 + z) x , where z is an element of C p with |z| < 1, we have This is immediate, since the function φ has nth Mahler coefficient We now define a C p -valued pairing on the C p -vector space C(Z p , C p ) by The following lemma will be used several times.Proof.Bilinearity is clear.For symmetry, we observe that if l is a non-negative integer, then and this last expression is equal to ψ, φ by a reversal of steps (4.2)-(4.4)with the roles of φ and ψ interchanged.We show that the interchanging of the summation signs at (4.5) is permissible.By [9, Chap.2, Sect.1.2, Cor.], it is enough to show that, for any ǫ > 0, there are only finitely many pairs (k, l) such that |η(k, l)| > ǫ, where For this, we simply invoke Lemma 4.1.
In fact, we have the following more symmetric version of the double sum for φ, ψ appearing in the proof of Proposition 4.2.It is more symmetric because the order of summation does not privilege either direction, k or l, in the lattice of pairs (k, l); rather, it runs through the lattice diagonally.Proof.
We claim that this sequence converges to zero.Let ǫ > 0. We again use Lemma 4.
Corollary 4.7 If r, s ∈ B <1 (1, C p ), and if (γ r,n ) n≥0 is the sequence of Mahler coefficients of γ r , then Proof.On the one hand, On the other hand, Now use Corollary 4.6.
More generally, we have the following.

Convolution of Mahler coefficients
, where a n → 0 and b n → 0. Then We show that the sequence of elements  Note that, because (C p [[t]] 0 , +, •) is a commutative C p -algebra with no zero divisors, the same is true of (C(Z p , C p ), +, ⋆).

Now consider the mutually inverse maps
We will also require the following notation.Proof.Let the Mahler coefficients of φ and ψ be (a k ) k≥0 and (b l ) l≥0 respectively.Then by (5.3), But this is equal to φ, ψ by Proposition 4.3.
, where 1 is the function with constant value 1.
Proof.Proof.Suppose that φ = ∞ k=0 a k β k and ψ = ∞ l=0 b l β l .On the one hand, by Proposition 2.2, while on the other hand Note that we have set a −1 = b −1 = 0. We have to show that

An integral transform
Recall the notation φ ⋆m defined in (5.4) and (5.5) for φ ∈ C(Z p , C p ) and m ∈ Z.
Given ψ ∈ C(Z p , C p ), we may consider the function We show that this function, defined initially on Z, can be extended to a locally analytic function Z p → C p .

Iterates of
Proof.We proceed by induction on m, the case m = 0 being vacuous.Let m ≥ 0, and assume that the statement is true for that m, i.e., Then where in the last step we have used a standard binomial identity and reincorporated the k = 0 and k = m + 1 terms.The induction is complete.
If we take ψ = γ r in Corollary 6.2, then we obtain If ψ ∈ C(Z p , C p ), then ψ(Z p ) is bounded, so the sequence (n! ψ(−1 − n)) n≥0 converges to zero, and so we have a well-defined, continuous function that is, Proof.We apply Proposition 2.6 (in the a ′ n notation).By the boundedness of ψ(Z p ), we may choose and we are done.
Remark.The map I : C(Z p , C p ) → la(Z p , C p ) is injective.Indeed, I (ψ) determines its own Mahler coefficients and therefore the values ψ(−1 − n) for all n ≥ 0. By continuity, these values determine the function ψ.
has a unique extension to a continuous function Z p → C p .This continuous extension is in fact I (ψ) and is therefore locally analytic on Z p .
Proof.Uniqueness is clear by continuity, so we turn immediately to existence.By Corollary 6.2, if m ∈ Z ≥0 , then In summary, It remains to show that the equality S m (ψ)(−1) = I (ψ)(m) holds for m ∈ Z <0 as well.

Manifesting I as an integral transform
We now turn to our claim that the function m → Zp (1 − β 1 ) ⋆m dµ(H ψ ) in (6.1) can be extended to a locally analytic function on Z p , showing at the same time that this extension is simply I (ψ).We hence derive an integral-transform expression for p-adic incomplete Γ-functions.Define where, as always, β n (x) = x n .Note that, for any fixed s ∈ Z p , the function ∞ n=0 (−1) n s n n! β n is indeed continuous, because its sequence of Mahler coefficients tends to zero.
Let • M denote the norm on C(Z p , C p ) defined by φ M = max n≥0 |a n |, where (a n ) n≥0 is the sequence of Mahler coefficients of φ.Proposition 6.8 The map T is uniformly continuous with respect to and for n ≥ 1, If s ∈ Z p , we define (1 − β 1 ) ⋆s to be T (s).By Proposition 6.9, this definition extends (1 − β 1 ) ⋆s from s ∈ Z to s ∈ Z p .Theorem 6.10 Let ψ ∈ C(Z p , C p ), and let I (ψ) be the locally analytic function defined in (6.2).Then for all s ∈ Z p , We first show that Int(ψ) is continuous.Write H ψ (t) = ∞ n=0 b n t n , and let N = sup n≥0 |b n |.Note that the supremum is finite because ψ is bounded.If ǫ > 0, then by Proposition 6.8, there is δ > 0 such that T (s 1 )−T (s 2 ) M < ǫ/N whenever |s 1 − s 2 | < δ.Take such s 1 , s 2 , and write T (s 1 ) − T (s 2 ) = ∞ n=0 a n β n , where a n → 0. Then This proves continuity.so because both I (ψ) and Int(ψ) are continuous, they are equal as functions on Z p .We thus have (6.8), and then (6.9) follows by Proposition 5.3.
Remark.We observe a similarity between the integral transform and the Laplace transform The equation I (Σψ) = −S −1 (I (ψ)) satisfied by the integral transform I (see Proposition 6.7) is akin to the equation L{tf (t)} = −L{f } ′ satisfied by the Laplace transform.In our situation, the derivative operator is replaced by S −1 , and the multiplication-by-t map is replaced by the Σ operator.We will also see in Section 6.4 an analogue of the Laplace-transform formula L{f * g} = L{f }L{g}.We compare the above formula with the definition of the complex incomplete Γ-function:

Image of I
We determine which functions are in the image Im(I ) of the injective map I : C(Z p , C p ) → la(Z p , C p ).The following proposition will help in the task.Proposition 6.13 If (ψ k ) k≥0 is a sequence of functions in C(Z p , C p ) converging uniformly to some ψ, then the sequence of functions I (ψ k ) converges uniformly to I (ψ).
Proof.The first assertion is clear by Proposition 6.15.As for the uniqueness of the coefficients for a given Ψ, if ∞ n=0 a n ∇ n q = 0, so ψ = 0 by the injectivity of I , and therefore the a n are all zero.
Corollary 6.17For all n ≥ 0, 1 n! ∇ n q = S −n (q).Therefore, Im(I ) consists of the functions where (a n ) n≥0 is a sequence in C p converging to zero.Again, the a n are determined uniquely by Ψ.
Proof.We first recall Proposition 6.7, which, once iterated, says that I (Σ n ψ) = (−1) n S −n (I (ψ)).We apply this in the case where ψ = β 0 .Then because β n = Σ n β 0 , we have the last equality by Proposition 6.15.On the other hand, I (β n ) = (−1) n 1 n! ∇ n q by Proposition 6.15 again.Therefore, S −n (q) = 1 n! ∇ n q, so the rest of the statement to be proven follows from Corollary 6.16.
We note in passing that We make another observation regarding the image of I .

I and convolution
Recall the convolution product ⋆ on C(Z p , C p ).We define a similar product on Im(I ), again via the algebra C p [[t]] 0 (see the beginning of Section 5).Consider the mutually inverse maps b n S −n (q).(6.12) These maps are well defined by Corollary 6.17.Denote by ⋄ the product on Then (Im(I ), +, ⋄) is a commutative C p -algebra with no zero divisors.Proof.Let the Mahler coefficients of φ and ψ be (b k ) k≥0 and (c l ) l≥0 respectively.Then Proposition 6.19 is akin to the convolution formula for the Laplace transform.
7 Connection to the equation We describe the role of the automorphism S in the solution to the well-known differential equation ] is a given power series with bounded coefficients.Of course, our intention is not to attempt to contribute anything to the general theory of p-adic differential equations, which has a vast literature extending far beyond this equation, but simply to highlight a connection.
It is worth remarking that, trivial though the equation F ′ + F = G appears to be, it is not at first sight obvious that if G ∈ Z p [[t]], say, then the equation has a solution F that is also in Z p [[t]] (which it does).
We have already seen that if ] has coefficients given by a continuous function ψ : Z p → C p , then F ′ + F = G where ψ being the function x → ψ(−1 − x).Just replace ψ by ψ in Proposition 3.1.A similar fact holds even when the coefficients of G are not assumed to vary continuously with n and are instead assumed only to be bounded.We turn now to this generalization and show how the automorphism S still appears.
Let K be a complete subfield of C p .For a positive real number M , we define F M (Z p , K) to consist of the functions ψ : Hence, remembering that J ψ (t) = (1 − t)G(t), we have J ′ φ (t) + J φ (t) = G(t).
r)) for all n ≥ 1.Thus, the p-adic locally analytic function γ r interpolates the values τ p (Γ(n, r)) at positive integers n.The existence of a p-adic locally analytic function with this interpolation property is what O'Desky and Richman showed, but via a different method.Of course, since g 1 = 1, we have γ 1 = q.
− y| for all x, y ∈ Z p .Such a function is obviously continuous.Proposition 2.12 Let A = {x ∈ C p | |x| ≤ 1}.The map S restricts to an automorphism of the A-module of 1-Lipschitz functions Z p → A.

Lemma 4 . 1 Proposition 4 . 2
Let (a k ) k≥0 and (b l ) l≥0 be sequences in C p that converge to zero.Then for any ǫ > 0, there areM, N ≥ 0 such that if either k ≥ M or l ≥ N , then |a k b l | ≤ ǫ.Proof.Fix ǫ > 0. First, choose L ≥ 0 such that |b l | ≤ 1 for all l ≥ L, possible because b l → 0 by assumption.Then, for each l < L, choose M l ≥ 0 such that |a k | |b l | ≤ ǫ for all k ≥ M l , which is again possible because a k → 0. Next, choose M such that |a k | ≤ ǫ for k ≥ M .Now let M = max{ M , M 0 , . . ., M L−1 }.Suppose that k ≥ M , and let l ≥ 0 be arbitrary.If l < L, then because k ≥ M ≥ M l , we have |a k | |b l | ≤ ǫ.Otherwise, if l ≥ L, then |b l | ≤ 1, while |a k | ≤ ǫ because k ≥ M , so |a k | |b l | ≤ ǫ.In summary, M has the property that |a k b l | ≤ ǫ whenever k ≥ M , irrespective of the value of l.One may prove similarly the existence of an N such that |a k b l | ≤ ǫ whenever l ≥ N , irrespective of the value of k.The pairing •, • is a non-degenerate symmetric bilinear form.
so it is enough to show that there are M, N ≥ 0 such that (i) for all k ≥ M and all l ≥ 0, |a k | |b l | ≤ ǫ, and (ii) for all k ≥ 0 and all l ≥ N , |a k | |b l | ≤ ǫ.

Corollary 4 . 6
If r, s ∈ B <1 (1, C p ), then g r , γ s = γ r , g s , where the functions γ r , g r are as in Section 2.5.

Corollary 4 . 8
If r ∈ B <1 (1, C p ) and (γ r,n ) n≥0 is the sequence of Mahler coefficients of γ r , then for any ψ ∈ C(Z p , C p ), ∞ n=0 n−k converges to zero as n → ∞.Let ǫ > 0. By Lemma 4.1 again, we may choose M, N ≥ 0 such that if either k ≥ M or l ≥ N , then |a k b l | ≤ ǫ.If n ≥ M + N and 0 ≤ k ≤ n, then either k ≥ M or n − k ≥ N , so |a k b n−k | ≤ ǫ, and then n k=0 n k a k b n−k ≤ ǫ.
Proposition 2.6 If f : Z p → C p has Mahler coefficients (a n ) n , then f is locally analytic on Z p if and only if there are β > 0 and N ∈ Z ≥0 such that v p (a n ) ≥ nβ for all n ≥ N .Equivalently, if we set a ′ n = a n /n!, then f is locally analytic on Z p if and only if there are .2) They are C p -linear maps but not algebra homomorphisms if C(Z p , C p ) has the usual product.We transfer the product on C p [[t]] 0 ⊆ C[[t]] to a new product ⋆ on C(Z p , C p ) via the above maps.Concretely, if φ and ψ have Mahler coefficients (a k ) k≥0 and (b l ) l≥0 respectively, then p , C p ) via (5.1) and (5.2) is exp(−t)G * ,φ (t).