Research Article | Open Access
A Recurrence Formula for Numbers
we establish a recurrence formula for numbers . A generating function for numbers is also presented.
1. Introduction and Results
where , with being the set of positive integers.
The numbers satisfy the recurrence relation (see )
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.
Proof of Theorem 1.2. Note the identity (see )
where . We have
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.
This work was Supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).
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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.