New recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials

In particular, the rational numbers Bn = Bn() and integers En = En(/) are called the classical Bernoulli numbers andEuler numbers, respectively. These polynomials andnumbers play important roles in various branches of mathematics including number theory, combinatorics, special functions and analysis, and there exist numerous interesting properties for them, see, for example, [–]. In , Agoh and Dilcher [] made use of some connections between the classical Bernoulli numbers and the Stirling numbers of the second kind to establish a quadratic recurrence formula on the classical Bernoulli numbers, which was generalized to the classical Bernoulli polynomials by He and Zhang []. More recently, He andWang [] extended the Agoh and Dilcher’s quadratic recurrence formula on the classical Bernoulli numbers to the Apostol-Bernoulli and Apostol-Euler polynomials. As further applications, they derived some corresponding results related to some formulae of products of the classical Bernoulli and Euler polynomials and numbers stated in Nielsen’s book [].


Introduction
The classical Bernoulli polynomials B n (x) and Euler polynomials E n (x) are usually defined by means of the following generating functions In particular, the rational numbers B n = B n () and integers E n =  n E n (/) are called the classical Bernoulli numbers and Euler numbers, respectively. These polynomials and numbers play important roles in various branches of mathematics including number theory, combinatorics, special functions and analysis, and there exist numerous interesting properties for them, see, for example, [-].
In , Agoh and Dilcher [] made use of some connections between the classical Bernoulli numbers and the Stirling numbers of the second kind to establish a quadratic recurrence formula on the classical Bernoulli numbers, which was generalized to the classical Bernoulli polynomials by He and Zhang []. More recently, He and Wang [] extended the Agoh and Dilcher's quadratic recurrence formula on the classical Bernoulli numbers to the Apostol-Bernoulli and Apostol-Euler polynomials. As further applications, they derived some corresponding results related to some formulae of products of the classical Bernoulli and Euler polynomials and numbers stated in Nielsen's book []. http://www.advancesindifferenceequations.com/content/2013/1/247 We begin by recalling now the Apostol-Bernoulli polynomials B (α) n (x; λ) and Apostol-Euler polynomials E (α) n (x; λ) of (real or complex) higher order α, which were introduced by Luo

Recurrence formulae for Apostol-Bernoulli polynomials
In what follows, we shall always denote by δ ,λ the Kronecker symbol given by δ ,λ =  or , according to λ =  or λ = , and we also denote by β n (x; λ) = B n+ (x; λ)/(n + ) for any nonnegative integer n. We first state the following.
Theorem . Let k, m, n be any non-negative integers. Then Proof Multiplying both sides of the identity Since B  (x; λ) =  when λ =  and B  (x; λ) =  when λ =  (see, e.g., []), then by setting More generally, the Taylor theorem gives Hence, applying (.) and (.) to (.), we get (  .  ) http://www.advancesindifferenceequations.com/content/2013/1/247 where M is denoted by In view of (.), we have , in view of changing the order of the summation, we obtain (  .   ) http://www.advancesindifferenceequations.com/content/2013/1/247 So from (.), (.), (.) and the symmetric distributions for the Apostol-Bernoulli poly- for non-negative integer k, then the identity above can be rewritten as It follows from (.) that Thus, combining (.) and (.), and then making k-times derivative with respect to v, we get which together with the Cauchy product arises the desired result after comparing the coefficients of u m v n /m!n!.
It follows that we show a special case of Theorem .. We have the following formula of products of the Apostol-Bernoulli polynomial due to He and Wang [].

Corollary . Let m, n be any positive integers. Then
(  .   ) http://www.advancesindifferenceequations.com/content/2013/1/247 Note that for any negative integers i, n, It follows from (.) and (.) that Thus, replacing x by yx and y by x in (.) gives the desired result.
We now use Theorem . to give another new recurrence formula for the Apostol-Bernoulli polynomials.
Theorem . Let k, m, n be any non-negative integers. Then for x + y + z = , We shall use induction on k. Clearly, (.) holds trivially when k =  in Theorem .. Now assume (.) for any smaller value of k. In light of (.), we have Since (.) holds for any smaller value of k then (.) http://www.advancesindifferenceequations.com/content/2013/1/247 Note that for non-negative integers i, k, m, n, by using induction on k. It follows from (.)-(.) that Hence, applying the five identities (.)-(.) above to (.), and then combining (.), gives (.). Thus, by setting x + y + z =  in (.), and using the symmetric distributions of the Apostol-Bernoulli polynomials, we complete the proof of Theorem . by replacing n by k, k by m and m by n.
We now give a special case of Theorem .. We have the following quadratic recurrence formula for the Apostol-Bernoulli polynomials, presented in [].

Recurrence formulae for mixed Apostol-Bernoulli and Apostol-Euler polynomials
We next give a similar formula to Theorem ., which is involving the mixed Apostol-Bernoulli and Apostol-Euler polynomials. As in the proof of Theorem ., we need the following formula concerning the mixed Apostol-Bernoulli and Apostol-Euler polynomials.

Theorem . Let k, m, n be any non-negative integers. Then
Proof Multiplying both sides of the identity (  .  ) http://www.advancesindifferenceequations.com/content/2013/1/247 In a similar consideration to (.), we have So from the symmetric distributions for the Apostol-Euler polynomials λE n ( -x; λ) = (-) n E n (x;  λ ), n ≥  (see, e.g., []), we get Applying (.) and (.) to (.), and then making k-times derivative with respect to v, we obtain which together with the Cauchy product gives the desired result by comparing the coefficients of u m v n /m!n!.
It follows that we show a special of Theorem .. We have the following formulae of products of the Apostol-Euler polynomials due to He and Wang []. http://www.advancesindifferenceequations.com/content/2013/1/247

Corollary . Let m, n be any non-negative integers. Then
Thus, the desired result follows by replacing x by yx and y by x in (.).
Now we apply Theorem . to give the following another recurrence formula for the mixed Apostol-Bernoulli and Apostol-Euler polynomials.
Theorem . Let k, m, n be any non-negative integers. Then for x + y + z = , Proof We firstly prove that for any non-negative integers k, m, n, (  .   ) http://www.advancesindifferenceequations.com/content/2013/1/247 The proof is similar to that of (.), and, therefore, we leave out some of the more obvious details. Clearly, the case k =  in (.) is complete. Next, consider the case k ≥  in (.). Assume that (.) holds for all positive integers being less than k. In light of (.), we have It follows from (.) and (.) that Hence, applying (.), (.) and (.) to (.), we conclude the induction step. Thus, by setting x + y + z =  in (.), and applying the symmetric distributions of Apostol-Euler polynomials, we complete the proof of Theorem . by replacing n by k, k by m and m by n.
We next give some special cases of Theorem .. We have the following formula products of the mixed Apostol-Bernoulli and Apostol-Euler polynomials due to He and Wang []. Proof Setting k = , and substituting  -y for y, yx for z, λ for μ and  λμ for λ in Theorem ., by the symmetric distributions of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result follows immediately.
We next apply Theorem . to give the following quadratic recurrence formulae for the mixed Apostol-Bernoulli and Apostol-Euler polynomials presented in []. Proof Substituting μ for λ,  λμ for μ, x for y and y for z in Theorem ., by applying the symmetric distributions of the Apostol-Euler polynomials, the desired result follows immediately.
Remark . We also mention that Theorem . above can also be used to obtain the formulae of products of the mixed Apostol-Bernoulli and Apostol-Euler polynomials. For example, setting m = , and substituting yx for x, x for y, n for k and m for n in Theorem ., with the help of the symmetric distributions of the Apostol-Bernoulli polynomials, Corollary . follows immediately. For some corresponding applications of Corollaries ., ., . and ., one is referred to [, , ].