IJPAM: Volume 53, No. 1 (2009)

GENERALIZED DIFFERENCE OPERATOR OF THIRD KIND
AND ON SECOND PARTIAL SUMS OF PRODUCTS
OF CONSECUTIVE TERMS OF ARITHMETIC AND
ARITHMETICO-GEOMETRIC PROGRESSIONS

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 $\Delta_{\ell_1,\ell_2}$ to third kind $\Delta_{\ell_1,\ell_2,\ell_3}$ for the positive reals $\ell_1,\ell_2$ and $\ell_3$ by presenting some results on generalized polynomial factorials of several kinds, generalized Leibnitz Theorem and Newton's formula. Also we develop a formula to find second partial sums of products of $n$ consecutive terms of arithmetic, arithmetico-geometric progressions by defining the inverse operator $\Delta^{-1}_{\ell_1,\ell_2,\ell_3}$ of the third kind operator $\Delta_{\ell_1,\ell_2,\ell_3}$.

Received: April 15, 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