IJPAM: Volume 53, No. 1 (2009)

ON GENERALIZED DIFFERENCE OPERATOR OF THIRD
KIND AND ITS APPLICATIONS IN NUMBER THEORY

M. Maria Susai Manuel$^1$, G. Britto Antony Xavier$^2$, V. Chandrasekar$^3$,
R. Pugalarasu$^4$, S. Elizabeth$^5$
$^{1,2,3,4}$Department of Mathematics
Sacred Heart College
Tirupattur, 635 601, Tamil Nadu, INDIA
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 $\Delta_\ell$ and the second kind $\Delta_{\ell_1,\ell_2}$ to the generalized difference operator of the third kind $\Delta_{\ell_1,\ell_2,\ell_3}$ for the positive reals $\ell_1,\ell_2$ and $\ell_3$. We also present the discrete version of Leibnitz Theorem, binomial theorem, Newton's formula with reference to $\Delta_{\ell_1,\ell_2,\ell_3}.$ Also by defining its inverse, we establish a few formulae for the sum of the second partial sums of higher powers of arithmetic progression in number theory.

Received: March 30, 2009

AMS Subject Classification: 39A

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: 53
Issue: 1