#### Abstract

Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials are established by applying the generating function methods and some summation transform techniques, and various known results are derived as special cases.

#### 1. Introduction

The classical Bernoulli polynomials and Euler polynomials are usually defined by means of the following generating functions: In particular, and are called the classical Bernoulli numbers and Euler numbers, respectively. These numbers and polynomials play important roles in many branches of mathematics such as combinatorics, number theory, special functions, and analysis. Numerous interesting identities and congruences for them can be found in many papers; see, for example, [1–4].

Some analogues of the classical Bernoulli and Euler polynomials are the Apostol-Bernoulli polynomials and Apostol-Euler polynomials . They were respectively introduced by Apostol [5] (see also Srivastava [6] for a systematic study) and Luo [7, 8] as follows: Moreover, and are called the Apostol-Bernoulli numbers and Apostol-Euler numbers, respectively. Obviously and reduce to and when . Some arithmetic properties for the Apostol-Bernoulli and Apostol-Euler polynomials and numbers have been well investigated by many authors. For example, in 1998, Srivastava and Todorov [9] gave the close formula for the Apostol-Bernoulli polynomials in terms of the Gaussian hypergeometric function and the Stirling numbers of the second kind. Following the work of Srivastava and Todorov, Luo [7] presented the close formula for the Apostol-Euler polynomials in a similar technique. After that, Luo [10] obtained some multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials. Further, Luo [11] showed the Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials by applying the Lipschitz summation formula and derived some explicit formulae at rational arguments for these polynomials in terms of the Hurwitz zeta function.

In the present paper, we will further investigate the arithmetic properties of the Apostol-Bernoulli and Apostol-Euler polynomials and establish some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials by using the generating function methods and some summation transform techniques. It turns out that various known results are deduced as special cases.

#### 2. The Restatement of the Results

For convenience, in this section we always denote by the Kronecker symbol given by or 1 according to or , and we also denote by the maximum number of the real numbers and by the maximum integer less than or equal to the real number . We now give the formula of products of the Apostol-Bernoulli polynomials in the following way.

Theorem 2.1. *Let and be any positive integers. Then,
*

* Proof. *Multiplying both sides of the identity
by , we obtain
It follows from (2.3) that
By the Taylor theorem we have
Since when and when (see e.g., [8]), by (1.2) and (2.5) we get
Putting (1.2) and (2.6) in (2.4), with the help of the Cauchy product, we derive
If we denote the left-hand side of (2.7) by and
then applying (1.2) to (2.8), in light of (2.7), we have
On the other hand, a simple calculation implies when and
when . Applying to (1.2), in view of changing the order of the summation, we obtain
It follows from (1.2), (2.10), (2.11), and the symmetric relation for the Apostol-Bernoulli polynomials for any nonnegative integer (see e.g., [8]) that
Thus, by equating (2.9) and (2.12) and then comparing the coefficients of , we complete the proof of Theorem 2.1 after applying the symmetric relation for the Apostol-Bernoulli polynomials.

It follows that we show some special cases of Theorem 2.1. By setting in Theorem 2.1, we have the following.

Corollary 2.2. *Let and be any positive integers. Then,
*

It is well known that the classical Bernoulli numbers with odd subscripts obey and for any positive integer (see, e.g., [12]). Setting in Corollary 2.2, we immediately obtain the familiar formula of products of the classical Bernoulli polynomials due to Carlitz [13] and Nielsen [14] as follows.

Corollary 2.3. *Let and be any positive integers. Then,
*

Since the Apostol-Bernoulli polynomials satisfy the difference equation = for any positive integer (see, e.g., [8]), by substituting for in Theorem 2.1 and then taking differences with respect to , we get the following result after replacing by .

Corollary 2.4. *Let and be any positive integers. Then,
*

Setting and in Corollary 2.4, by for any nonnegative integer , we get the following.

Corollary 2.5. *Let and be any positive integers. Then,
*

In particular, the case in Corollary 2.5 gives the following generalization for Woodcock’s identity on the classical Bernoulli numbers, see [15, 16],

Corollary 2.6. *Let and be any positive integers. Then,
*

We next present some mixed formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials and numbers.

Theorem 2.7. *Let and be non-negative integers. Then,
*

*Proof. *Multiplying both sides of the identity
by , we obtain
It follows from (2.20) that
Applying (1.3) and (2.6) to (2.21), in view of the Cauchy product, we get
On the other hand, since the left-hand side of (2.22) vanishes when , it suffices to consider the case . Applying to (1.3), in view of changing the order of the summation, we have
It follows from (1.3), (2.23), and the symmetric relation for the Apostol-Euler polynomials for any non-negative integer (see, e.g., [7]) that
Thus, by equating (2.22) and (2.24) and then comparing the coefficients of , we complete the proof of Theorem 2.7 after applying the symmetric relation for the Apostol-Euler polynomials.

Next, we give some special cases of Theorem 2.7. By setting in Theorem 2.7, we have the following.

Corollary 2.8. *Let and be non-negative integers. Then,
*

Since the classical Euler polynomials at zero arguments satisfy , , and for any positive integer (see, e.g., [12]), by setting in Corollary 2.8, we obtain the following.

Corollary 2.9. *Let and be non-negative integers. Then,
**
where when () and otherwise.*

Theorem 2.10. *Let be non-negative integer and positive integer. Then,
*

*Proof. *Multiplying both sides of the identity
by , we obtain
By (1.3) and the Taylor theorem, we have
Applying (1.2), (1.3), and (2.30) to (2.29), we get
which means
Thus, by comparing the coefficients of in (2.32), we conclude the proof of Theorem 2.10 after applying the symmetric relation for the Apostol-Euler polynomials.

Obviously, by setting in Theorem 2.10, we have the following.

Corollary 2.11. *Let be non-negative integer and positive integer. Then,
*

Since , , , and for any positive integer , by setting in Corollary 2.11, we obtain the following.

Corollary 2.12. *Let be non-negative integer and positive integer. Then,
*

*Remark 2.13. *For the equivalent forms of Corollaries 2.9 and 2.12, the interested readers may consult [14].

#### Acknowledgments

The authors are very grateful to anonymous referees for helpful comments on the previous version of this work. They express their gratitude to Professor Wenpeng Zhang who provided them with some suggestions. This paper is supported by the National Natural Science Foundation of China (Grant no. 10671194).