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

THEORY OF GENERALIZED DIFFERENCE OPERATOR
OF -TH KIND AND ITS APPLICATIONS
IN NUMBER THEORY (PART - I)

R. Pugalarasu, M. Maria Susai Manuel,
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.