Research Article | Open Access

Volume 2009 |Article ID 605313 | https://doi.org/10.1155/2009/605313

Guodong Liu, "A Recurrence Formula for Numbers ", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 605313, 6 pages, 2009. https://doi.org/10.1155/2009/605313

# A Recurrence Formula for 𝐷 Numbers 𝐷(2𝑛−1)2𝑛

Accepted16 Oct 2009
Published19 Nov 2009

#### 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 )

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 Nörlund numbers (see [2, 4, 6]). A generating function for the Nörlund 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 )

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

The numbers are called the -Nörlund numbers that satisfy the recurrence relation (see )

so we find

A generating function for the -Nörlund numbers is (see )

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 -Nörlund numbers :

Proof of Theorem 1.2. Note the identity (see ) 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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