ISRN Discrete Mathematics

Volume 2013 (2013), Article ID 810245, 5 pages

http://dx.doi.org/10.1155/2013/810245

Research Article

Some Properties of a Sequence Similar to Generalized Euler Numbers

Haiqing Wang and Guodong Liu

Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China

Received 27 December 2012; Accepted 7 February 2013

Academic Editors: W. F. Klostermeyer, T. Prellberg, S. Rim, and W. F. Smyth

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.

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 [1–4])

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 [8–11]) 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.

References

G. D. Liu, “On congruences of Euler numbers modulo powers of two,” Journal of Number Theory, vol. 128, no. 12, pp. 3063–3071, 2008.

G. D. Liu and H. M. Srivastava, “Explicit formulas for the Nörlund polynomials and ,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1377–1384, 2006.

G. D. Liu and W. P. Zhang, “Applications of an explicit formula for the generalized Euler numbers,” Acta Mathematica Sinica, vol. 24, no. 2, pp. 343–352, 2008.

Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York, NY, USA, 1969.

H. M. Srivastava and G. D. Liu, “Some identities and congruences involving a certain family of numbers,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 536–542, 2009.

Z. H. Sun, “Identities and congruences for a new sequence,” International Journal of Number Theory, vol. 8, no. 1, pp. 207–225, 2012.

Z. H. Sun, “Some properties of a sequence analogous to Euler numbers,” Bulletin of the Australian Mathematical Society, 2012.

C. Jordan, Calculus of Finite Differences, Chelsea, New York, NY, USA, 1965.

G. D. Liu, “The numbers and the central factorial numbers,” Publicationes Mathematicae Debrecen, vol. 79, no. 1-2, pp. 41–53, 2011.

G. D. Liu, “A recurrence formula for numbers ,” Discrete Dynamics in Nature and Society, Article ID 605313, 6 pages, 2009.

N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, Germany, 1924, Reprinted by Chelsea, Bronx, New York, NY, USA, 1954.

L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, The Netherlands, 1974, (Translated from the French by J. W. Nienhuys).

J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, NY, YSA, 1968.