International Scholarly Research Notices

International Scholarly Research Notices / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 810245 | 5 pages | https://doi.org/10.1155/2013/810245

Some Properties of a Sequence Similar to Generalized Euler Numbers

Academic Editor: W. F. Smyth
Received27 Dec 2012
Accepted07 Feb 2013
Published11 Mar 2013

Abstract

We introduce the sequence given by generating function and establish some explicit formulas for the sequence . Several identities involving the sequence , Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

1. Introduction and Definitions

For a real or complex parameter , the generalized Euler polynomials are defined by the following generating function (see [14]) Obviously, we have in terms of the classical Euler polynomials , being the set of positive integers. The classical Euler numbers are given by the following:

The so-called the generalized Euler numbers are defined by (see [3, 5]) In fact, are the Euler numbers of order , being the set of integers. The numbers are the ordinary Euler numbers.

Zhi-Hong Sun introduces the sequence similar to Euler numbers as follows (see [6, 7]): where (and in what follows) is the greatest integer not exceeding .

Clearly, for . The first few values of are shown below

The sequence is related to the classical Bernoulli polynomials (see [811]) and the classical Euler polynomials . Zhi-Hong Sun gets the generating function of and deduces many identities involving . As example, (see [6]),

Similarly, we can define the generalized sequence . For a real or complex parameter , the generalized sequence is defined by the following generating function: Obviously, By using (10), we can obtain

We now return to the Stirling numbers of the first kind, which are usually defined by (see [2, 5, 8, 11, 12]) or by the following generating function:

It follows from (13) or (14) that and that

The central factorial numbers are given by the following expansion formula (see [3, 5, 13]): or by means of the generating function

It follows from (17) or (18) that with We also find from (18) that

The main purpose of this paper is to prove some formulas for the generalized sequence and . Some identities involving the sequence , Stirling numbers , and the central factorial numbers are deduced.

2. Main Results

Theorem 1. Let and Then,

Remark 2. By (15), (19), (20), and Theorem 1, we know that is a polynomial of with integral coefficients. For example, by setting in Theorem 1, we get

Taking in Theorem 1, we can obtain the following.

Corollary 3. Let . Then,

From Corollary 3, we may immediately deduce the following results.

Corollary 4. Let . Then,

Theorem 5. Let . Then,

Theorem 6. Let . Suppose also that is defined by (22). Then,

Theorem 7. Let . Then,

Theorem 8. Let . Then,

Theorem 9. Let . Then,

3. Proofs of Theorems

Proof of Theorem 1. By (10), (13), and (18), we have which readily yields This completes the proof of Theorem 1.

Proof of Theorem 5. By (10), we have and , thus By Theorem 1 and comparing the coefficient of on both sides of (35), we get Again, by taking in Theorem 1, we have By (36) and (37), we immediately obtain (27). This completes the proof of Theorem 5.

Proof of Theorem 6. By applying Theorem 1, we have On the other hand, it follows from (10) that By using (38) and (39), we find that We now note that Hence, yields Comparing the coefficient of on both sides of (43), we immediately get (28). This completes the proof of Theorem 6.

Proof of Theorem 7. Consider Thus, By (42) and (45) we obtain (29). This completes the proof of Theorem 7.

Proof of Theorem 8. By using (7), we have Thus That is, Comparing the coefficient of on both sides of (48), we get the following: By (49) we immediately obtain (30). This completes the proof of Theorem 8.

Proof of Theorem 9. By integrating (7) with respect to from 0 to 1, we have By (50) and ( is constant), we have (31). This completes the proof of Theorem 9.

Acknowledgments

This work is partly supported by the Social Science Foundation (no. 2012YB03) of Huizhou University and the Key Discipline Foundation (no. JG2011019) of Huizhou University.

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Copyright © 2013 Haiqing Wang and Guodong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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