Abstract and Applied Analysis

Volume 2009, Article ID 430452, 7 pages

http://dx.doi.org/10.1155/2009/430452

## Some Computational Formulas for -Nörlund Numbers

Department of Mathematics, Huizhou University, Huizhou, Guangdong 516015, China

Received 30 June 2009; Accepted 11 October 2009

Academic Editor: Lance Littlejohn

Copyright © 2009 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

The author establishes some identities involving the numbers, Bernoulli numbers, and central factorial numbers of the first kind. A generating function and several computational formulas for -Nörlund numbers are also presented.

#### 1. Introduction and Results

The Bernoulli polynomials of order , for any integer , may be defined by (see [1–5])

The numbers are the Bernoulli numbers of order , are the ordinary Bernoulli numbers (see [2, 6, 7]). By (1.1), we can get (see [4, page 145]) where , with being the set of positive integers.

The numbers are called the Nörlund numbers (see [4, 8]). A generating function for the Nörlund numbers is (see [4, page 150])

The numbers may be defined by (see [4, 5])

By (1.1), (1.6), and note that (where ), we can get

Taking in (1.7), and note that , (see [4, pages 22 and 145]), we have

The numbers are called the -Nörlund numbers. These numbers and have many important applications. For example (see [4, page 246])

We now turn to the central factorial numbers of the first kind, which are usually defined by (see [9–12]) or by means of the following generating function:

It follows from (1.11) or (1.12) that and that where denotes the Kronecker symbol.

By (1.13), we have

The main purpose of this paper is to prove some identities involving numbers, Bernoulli numbers, and central factorial numbers of the first kind and obtain a generating function and several computational formulas for the -Nörlund numbers. That is, we will prove the following main conclusion.

Theorem 1.1. *Let , . Then
*

*Remark 1.2. *By (1.18), we may immediately deduce the following (see [4, page 147]:

Theorem 1.3. *Let . Then
*

*Remark 1.4. *By (1.20) and (1.17), we may immediately deduce the following:

Theorem 1.5. *Let . Then
**
so one finds **By (1.23), and note that**
one may immediately deduce the following Corollary 1.6.*

Corollary 1.6. *Let . Then
*

Theorem 1.7. *Let . Then*(i)*(ii)**(iii)*

Theorem 1.8. *Let . Then*(i)*(ii)*

#### 2. Proof of the Theorems

*Proof of Theorem 1.1. *By (1.4) and (1.3), we have
Setting in (2.1), we get
By (2.2) and (1.7), we immediately obtain (1.18). This completes the proof of Theorem 1.1.

*Proof of Theorem 1.3. *By the usage of Theorem 1.1 and (1.13).

*Proof of Theorem 1.5. *Note the identity (see [4, page 203])
we have
By (2.4) and (1.2), we have
that is,
By (2.6) and (1.7), we have
By (2.7) and (1.19), we immediately obtain (1.23). This completes the proof of Theorem 1.5.

*Proof of Theorem 1.7. *By (1.6), we have
where is an integer.

Setting in (2.8), and note that , we have

By (2.9), (1.19), (1.8), and (1.21), we immediately obtain (1.26).

Setting in (2.8), and note that , we have

By (2.10), (1.19), (1.8), and (1.21), we immediately obtain (1.27).

Setting in (2.8), and note that (1.20) and , we immediately obtain (1.18). This completes the proof of Theorem 1.7.

*Proof of Theorem 1.8. *Setting in (2.8), and note (1.19), (1.20), and (1.8), we immediately obtain (1.29).

Setting in (2.8), and note (1.22), (1.20), and (1.8), we immediately obtain (1.30). This completes the proof of Theorem 1.8.

#### Acknowledgment

This work was supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).

#### References

- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Higher Transcendental Functions*, vol. 1, McGraw-Hill, London, UK, 1953. 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 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 (Chinese). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. E. Nörlund,
*Vorlesungen über Differenzenrechnung*, Springer, Berlin, Germany, 1924, reprinted by Chelsea, Bronx, NY, USA, 1954. - F. R. Olson, “Some determinants involving Bernoulli and Euler numbers of higher order,”
*Pacific Journal of Mathematics*, vol. 5, pp. 259–268, 1955. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Liu and H. Luo, “Some identities involving Bernoulli numbers,”
*The Fibonacci Quarterly*, vol. 43, no. 3, pp. 208–212, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 MathSciNet - G. D. Liu, “Some computational formulas for Nörlund numbers,”
*The Fibonacci Quarterly*, vol. 45, no. 2, pp. 133–137, 2007. View at Google Scholar · View at MathSciNet - G. D. Liu, “Summation and recurrence formula involving the central factorial numbers and zeta function,”
*Applied Mathematics and Computation*, vol. 149, no. 1, pp. 175–186, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Liu, “Some identities involving the central factorial numbers and Riemann zeta function,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 5, pp. 715–725, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Liu, “The generalized central factorial numbers and higher order Nörlund Euler-Bernoulli polynomials,”
*Acta Mathematica Sinica. Chinese Series*, vol. 44, no. 5, pp. 933–946, 2001 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Riordan,
*Combinatorial Identities*, John Wiley & Sons, New York, NY, USA, 1968. View at MathSciNet