Research Article | Open Access

# A Recurrence Formula for Numbers

**Academic Editor:**Binggen Zhang

#### Abstract

we establish a recurrence formula for numbers . A generating function for numbers is also presented.

#### 1. Introduction and Results

The Bernoulli polynomials of order , for any integer , may be defined by (see [1β4])

The numbers are the Bernoulli numbers of order , are the ordinary Bernoulli numbers (see [2, 5]). By (1.1), we can get (see [4, page 145])

where , with being the set of positive integers.

The numbers are called the NΓΆrlund numbers (see [2, 4, 6]). A generating function for the NΓΆrlund numbers is (see [4, page 150])

The numbers may be defined by (see [4, 7, 8])

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

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

The numbers satisfy the recurrence relation (see [7])

By (1.9), we may immediately deduce the following (see [4, page 147]):

The numbers are called the -NΓΆrlund numbers that satisfy the recurrence relation (see [7])

so we find

A generating function for the -NΓΆrlund numbers is (see [7])

These numbers and have many important applications. For example (see [4, page 246])

The main purpose of this paper is to prove a recurrence formula for numbers and to obtain a generating function for numbers . That is, we will prove the following main conclusion.

Theorem 1.1. *Let . Then
**
so one finds *

Theorem 1.2. *Let be a complex number with . Then
*

#### 2. Proof of the Theorems

*Proof of Theorem 1.1. *Note the identity (see [4, page 203])
we have
Therefore,
By (2.3) and (1.2), we have
That is,
By (2.5) and (1.7), we have
Setting in (2.6), and note (1.10), we immediately obtain Theorem 1.1. This completes the proof of Theorem 1.1.

*Remark 2.1. *Setting in (2.6), and note (1.10), we may immediately deduce the following recurrence formula for -NΓΆrlund numbers :

*Proof of Theorem 1.2. *Note the identity (see [9])
where . We have
That is,

On the other hand,
Thus, by (2.10), (2.11), and Theorem 1.1, we have
That is,
By (2.13), and note that
we immediately obtain Theorem 1.2. This completes the proof of Theorem 1.2.

#### Acknowledgment

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

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#### Copyright

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.