Research Article | Open Access

Volume 2012 |Article ID 927953 | https://doi.org/10.1155/2012/927953

Yuan He, Chunping Wang, "Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials", Discrete Dynamics in Nature and Society, vol. 2012, Article ID 927953, 11 pages, 2012. https://doi.org/10.1155/2012/927953

# Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials

Revised13 Jul 2012
Accepted15 Jul 2012
Published09 Aug 2012

#### 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, .

Some analogues of the classical Bernoulli and Euler polynomials are the Apostol-Bernoulli polynomials and Apostol-Euler polynomials . They were respectively introduced by Apostol  (see also Srivastava  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  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  presented the close formula for the Apostol-Euler polynomials in a similar technique. After that, Luo  obtained some multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials. Further, Luo  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., ), 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., ) 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., ). Setting in Corollary 2.2, we immediately obtain the familiar formula of products of the classical Bernoulli polynomials due to Carlitz  and Nielsen  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., ), 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., ) 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., ), 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 .

#### 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).

1. Y. He and Q. Liao, “Some congruences involving Euler numbers,” The Fibonacci Quarterly, vol. 46/47, no. 3, pp. 225–234, 2008/09. View at: Google Scholar
2. Y. He and W. Zhang, “Some symmetric identities involving a sequence of polynomials,” Electronic Journal of Combinatorics, vol. 17, no. 1, Note 7, p. 7, 2010. View at: Google Scholar | Zentralblatt MATH
3. D. S. Kim, T. Kim, S.-H. Lee, D. V. Dolgy, and S.-H. Rim, “Some new identities on the Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 856132, 11 pages, 2011. View at: Publisher Site | Google Scholar
4. H. M. Srivastava and Á. Pintér, “Remarks on some relationships between the Bernoulli and Euler polynomials,” Applied Mathematics Letters, vol. 17, no. 4, pp. 375–380, 2004.
5. T. M. Apostol, “On the Lerch zeta function,” Pacific Journal of Mathematics, vol. 1, pp. 161–167, 1951. View at: Google Scholar | Zentralblatt MATH
6. H. M. Srivastava, “Some formulas for the Bernoulli and Euler polynomials at rational arguments,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 129, no. 1, pp. 77–84, 2000.
7. Q.-M. Luo, “Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 917–925, 2006. View at: Google Scholar | Zentralblatt MATH
8. Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 290–302, 2005.
9. H. M. Srivastava and P. G. Todorov, “An explicit formula for the generalized Bernoulli polynomials,” Journal of Mathematical Analysis and Applications, vol. 130, no. 2, pp. 509–513, 1988.
10. Q.-M. Luo, “The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order,” Integral Transforms and Special Functions, vol. 20, no. 5-6, pp. 377–391, 2009.
11. Q.-M. Luo, “Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials,” Mathematics of Computation, vol. 78, no. 268, pp. 2193–2208, 2009.
12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas Graphs, and Mathematical Tables,, vol. 55, National Bureau of Standards, Washington, DC, USA, 1964. View at: Zentralblatt MATH
13. L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol. 34, pp. 361–363, 1959.
14. N. Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, Paris, France, 1923.
15. Y. He and W. Zhang, “Some sum relations involving Bernoulli and Euler polynomials,” Integral Transforms and Special Functions, vol. 22, no. 3, pp. 207–215, 2011.
16. H. Pan and Z.-W. Sun, “New identities involving Bernoulli and Euler polynomials,” Journal of Combinatorial Theory A, vol. 113, no. 1, pp. 156–175, 2006.

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