# IJPAM: Volume 64, No. 1 (2010)

**THEORY OF GENERALIZED DIFFERENCE OPERATOR**

OF -TH KIND AND ITS APPLICATIONS

IN NUMBER THEORY (PART - I)

OF -TH KIND AND ITS APPLICATIONS

IN NUMBER THEORY (PART - I)

V. Chandrasekar, G. Britto Antony Xavier

Department of Mathematics

Sacred Heart College

Tirupattur, 635 601, Tamil Nadu, INDIA

Department of Science and Humanities

R.M.D. Engineering College

Kavaraipettai, 601 206, Tamil Nadu, INDIA

e-mail: manuelmsm_03@yahoo.co.in

**Abstract.**In this paper, we define the generalized difference operator of -th kind denoted as
for the real or complex valued function ,
as

and obtain its relation with and , the generalized difference and shift operators respectively. Also we present the discrete version of Leibnitz Theorem, binomial theorem and Newton's formula according to
. By defining the inverse,
and using 's, the Stirling numbers of the second kind, we establish a formula for the sum of
times partial sums of the -th powers of an arithmetic progression in number theory.

**Received: **July 18, 2010

**AMS Subject Classification: **39A12

**Key Words and Phrases: **generalized difference operator, generalized polynomial factorial, partial sums

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 64

**Issue:** 1