IJPAM: Volume 56, No. 1 (2009)

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

M. Maria Susai Manuel$^1$, V. Chandrasekar$^2$, G. Britto Antony Xavier$^3$,
R. Pugalarasu$^4$, S. Elizabeth$^5$
$^{1,2,3,4}$Department of Mathematics
Sacred Heart College
Tirupattur, Tamil Nadu, 635 601, INDIA
$^1$e-mail: [email protected]
$^{5}$Department of Mathematics
Auxilium College
Vellore, Tamil Nadu, INDIA


Abstract.In this paper, the authors extend the theory of the generalized difference operator $\Delta_\ell$ to the Generalized difference operator of the fourth kind $\Delta_{\ell_1,\ell_2,\ell_3,\ell_4}$ for the positive reals $\ell_1,\ell_2,\ell_3$ and $\ell_4$ and present the discrete version of Leibnitz Theorem, binomial theorem, Newton's formula with reference to $\Delta_{\ell_1,\ell_2,\ell_3,\ell_4}.$ Also by defining its inverse, we establish a few formulae for the sum of third partial sums $P^3S^n$, for $n=1,2,3...$ of the higher powers of the arithmetic progressions in number theory. Appropriate examples are provided to illustrate the results.

Received: August 17, 2009

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: 2009
Volume: 56
Issue: 1