Higher-Order Convolutions for Apostol-Bernoulli , Apostol-Euler and Apostol-Genocchi Polynomials

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.

Also the case λ = 1 in (4) gives the Bernoulli numbers B n , the Euler numbers E n and the Genocchi numbers G n as follows: Recently, the above-defined generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials was unified by the following generating function (see, for example, [5]): It is worth mentioning that the case α = 1 in (5) was constructed by Ozden et al. [6,7].It is easily seen that the polynomials Y n,β (x; κ, a, b) given by can be regarded as a generalization and unification of the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials with, of course, suitable choices of the parameter a, b and β.We refer to the recent works [8][9][10][11][12][13] on these Apostol-type polynomials and numbers.
In the present paper, we shall be concerned with some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials.The idea stems from the higher-order convolutions for the Bernoulli polynomials due to Agoh and Dilcher [14], Bayad and Kim [15] and Bayad and Komatsu [16].We establish several higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials by making use of the generating-function methods and summation-transform techniques.It turns out that several interesting known results are obtainable as special cases of our main results.
This paper is organized as follows.In Section 2, we first give the higher-order convolution for the polynomials defined by (5) Y n,β (x; κ, a, b) and then present the corresponding higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials.Moreover, several corollaries and consequences of our main theorems are also deduced.Section 3 is devoted to the proofs of the main results by applying the generating-function methods and summation-transform techniques.

Main Results
As usual, by ( λ n ) we denote the binomial coefficients given, for λ ∈ C, by We also denote by s(n, k) the Stirling numbers of the first kind and by S(n, k) the Stirling numbers of the second kind, which are usually defined by the following generating functions (see, for example, [17,18]): where 7) was first studied by Agoh and Dilcher [14] who proved an existence theorem and also derived some explicit expressions for k = 3 involving the Bernoulli polynomials.We now state the following higher-order convolution for the general Apostol-type polynomials Y n,β (x; κ, a, b) defined by (5).
Theorem 1.Let d be a positive integer and let Then, for an integer κ and for m, n ∈ N * , We first deduce some special cases of Theorem 1.By taking Thus, by applying (8) to Theorem 1, we get the following higher-order convolution for the Apostol-Euler polynomials.

Corollary 1. Let d be a positive integer and let
Obviously, in the case when m = 0, Corollary 1 yields the following further special case for d ∈ N and n ∈ N * : which, upon setting i → i + 1, corresponds to the following result for the Apostol-Euler polynomials due to Bayad and Kim [15] Theorem 4: If we change the order of the summation on the right-hand side of (9), we get In particular, upon setting λ = 1 in (10), we find for d ∈ N and n ∈ N * that (see, for example, ref. [19] Theorem 5) If we take α = κ = b = 1 in (5), we obtain the following relationships for n ∈ N * : Consequently, Theorem 1 can be applied in conjunction with (11) in order to obtain the corresponding higher-order convolutions for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials.
We proceed now to give here some much simpler expressions for the higher-order convolutions for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials.
Then, for m, n ∈ N * (m + n d), For λ = 1, Theorem 2 reduces to the following higher-order convolution for the Bernoulli polynomials: For a different expression than that given by ( 12) in its special case when see a known result [16] Corollary 4.
If we set m = 0 in Theorem 2, we get For r ∈ N and m, n ∈ N * , it is known that (see, for example, [20] Theorem 1.2) where λ n denotes the rising factorial of order n given by and with F(t) being a formal power series.Thus, by taking 15) and substituting n − d for m, i − 1 for n, y for x and 0 for y in (14), we find (for positive integers i, d, n with n d) that It is easily seen from the properties of the Beta function B(α, β) and the Gamma function Γ(z) that Let δ 1,λ be a Kronecker symbol given by Since B 0 (x; λ) = 1 when λ = 1 and B 0 (x; λ) = 0 when λ = 1 (see, for example, [3]), by setting in (16), with the help of (17), we have We find from ( 13) and ( 18) the following formula due to Bayad and Kim [15] Theorem 5 for sums of products of the Apostol-Bernoulli polynomials: Upon changing the order of the summation on the right-hand side of ( 13), we get which, in the special case when λ = 1, yields the following famous formula for the Bernoulli polynomials due to Dilcher [19] Theorem 3: Let p n,m (x) denote a polynomial given by (see, for example [21,22]) Then, by applying ( 20) and ( 22), we get which is a generalization of the following result given by Kim and Hu [22] Theorem 1.2 for the Apostol-Bernoulli numbers: Then, for m, n ∈ N * (m + n d), In its special case when m = 0, Theorem 3 immediately yields By a similar consideration to that for (19), we can obtain the following formula for the Apostol-Genocchi polynomials: By changing the order of the summation on the right-hand side of ( 23), we find that Finally, upon setting λ = 1 in (24), gives a formula for sums of products of the Genocchi polynomials, which is analogous to (21).

Proofs of Theorems
Before giving the proofs of Theorems 1-3, we recall the following auxiliary results which will be needed in our proofs.
Furthermore, for n ∈ N, Moreover, for r ∈ N, where and the sequence { f n (x)} ∞ n=0 is given as in Equation (15).
Proof of Theorem 1.First of all, by setting α = 1 in (25), we get Let ν ∈ N and let the function f ν (t) be differentiable with respect to t.If we set then it is clear from (5) that for l ∈ N * , By differentiating both sides of (28) m times with respect to t, with the help of the general Leibniz rule presented in [18] (pp.130-133), we obtain We now denote by [t n ] f (t) the coefficient of t n in f (t) for n ∈ N * .Then, by making use of the operation t n n! on both sides of (30) in conjunction with (29), we find that Also, by using the Leibniz rule, we have It follows from the above two identities that If we replace F(y, t) in (26) by and making use of ( 17), we find for n ∈ N * that  Proof of Theorem 3. From ( 11) and (31), we find for d ∈ N and m, n ∈ N * that • s(d, i).