ISRN Discrete Mathematics

Volume 2013, Article ID 810245, 5 pages

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

## Some Properties of a Sequence Similar to Generalized Euler Numbers

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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Liu and H. M. Srivastava, “Explicit formulas for the Nörlund polynomials ${B}_{n}^{(x)}$ and ${b}_{n}^{(x)}$,”
*Computers & Mathematics with Applications*, vol. 51, no. 9-10, pp. 1377–1384, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. L. Luke,
*The Special Functions and Their Approximations, Vol. I*, Academic Press, New York, NY, USA, 1969. View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. H. Sun, “Identities and congruences for a new sequence,”
*International Journal of Number Theory*, vol. 8, no. 1, pp. 207–225, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z. H. Sun, “Some properties of a sequence analogous to Euler numbers,”
*Bulletin of the Australian Mathematical Society*, 2012. View at Publisher · View at Google Scholar - C. Jordan,
*Calculus of Finite Differences*, Chelsea, New York, NY, USA, 1965. View at Zentralblatt MATH · View at MathSciNet - G. D. Liu, “The $D$ numbers and the central factorial numbers,”
*Publicationes Mathematicae Debrecen*, vol. 79, no. 1-2, pp. 41–53, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Liu, “A recurrence formula for $D$ numbers ${D}_{2n}^{(2n-1)}$,”
*Discrete Dynamics in Nature and Society*, Article ID 605313, 6 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - 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). View at Zentralblatt MATH · View at MathSciNet - J. Riordan,
*Combinatorial Identities*, John Wiley & Sons, New York, NY, YSA, 1968. View at MathSciNet