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

**THEORY OF GENERALIZED DIFFERENCE OPERATOR**

OF -TH KIND AND ITS APPLICATIONS

IN NUMBER THEORY (PART - II)

OF -TH KIND AND ITS APPLICATIONS

IN NUMBER THEORY (PART - II)

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 extend the theory of generalized difference operator of the -th kind,
, where 's are real and present the discrete version of Leibnitz Theorem and Newton's Formula with reference to
. Using the Stirling numbers of the second kind 's we establish a formula for the sum of the general partial sums of consecutive terms of an arithmetic progression and sum of the general partial sums of an arithmetico-geometric progression in number theory. Suitable examples are provided to illustrate the main results.

**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