IJPAM: Volume 56, No. 1 (2009)

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

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.As the theory of the generalized difference operator of first kind $\Delta_\ell$, second $\Delta_{\ell_1,\ell_2}$ and third $\Delta_{\ell_1,\ell_2,\ell_3}$ have been developed in [#!brit06!#], [#!mbc!#], [10], [#!mbc02!#], in this paper, the authors extend the theory of $\Delta_\ell$ to the fourth kind operator $\Delta_{\ell_1,\ell_2,\ell_3,\ell_4}$ for the positive reals $\ell_1,\ell_2,\ell_3$ and $\ell_4$ by presenting some results on generalized polynomial factorials of third and fourth kinds, generalized Leibnitz Theorem, Newton's formula and formula to find third partial sums of products of $n$ consecutive terms of arithmetic and arithmetic-geometric progression by defining its inverse operators $\Delta^{-1}_{\ell_1,\ell_2,\ell_3,\ell_4}$.

Received: August 18, 2009

AMS Subject Classification: 39A12

Key Words and Phrases: generalized difference operator, generalized factorial, partial sums

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 1