Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 430452 | 7 pages | https://doi.org/10.1155/2009/430452

Some Computational Formulas for -Nörlund Numbers

Academic Editor: Lance Littlejohn
Received30 Jun 2009
Accepted11 Oct 2009
Published18 Nov 2009

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 [15])

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 [912]) 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

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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.


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