IJPAM: Volume 64, No. 1 (2010)

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

R. Pugalarasu$^1$, M. Maria Susai Manuel$^2$,
V. Chandrasekar$^3$, G. Britto Antony Xavier$^4$
$^{1,3,4}$Department of Mathematics
Sacred Heart College
Tirupattur, 635 601, Tamil Nadu, INDIA
$^2$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 $n$-th kind, $\Delta_{\ell_1,\ell_2,\ldots,\ell_n}$, where $\ell_i$'s are real and present the discrete version of Leibnitz Theorem and Newton's Formula with reference to $\Delta_{\ell_1,\ell_2,\ldots,\ell_n}$. Using the Stirling numbers of the second kind $S_i^n$'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