IJPAM: Volume 64, No. 1 (2010)

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

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 define the generalized difference operator of $n$-th kind denoted as $\Delta_{\ell_1,\ell_2, \cdots \ell_n}$ for the real or complex valued function $u(k)$, $k\in [0, \infty)$ as
\begin{multline}
\Delta_{\ell_1,\ell_2, \cdots \ell_n} u(k) = u(k+\ell_1+\ell_2...
...-1)^{n-1}\big\{u(k+\ell_1)+\cdots+u(k+\ell_n)\big\}+(-1)^n u(k).
\end{multline}
and obtain its relation with $\Delta_{\ell}$ and $E$, the generalized difference and shift operators respectively. Also we present the discrete version of Leibnitz Theorem, binomial theorem and Newton's formula according to $\Delta_{\ell_1,\ell_2, \cdots \ell_n}$. By defining the inverse, $\Delta^{-1}_{\ell_1,\ell_2, \cdots \ell_n}$ and using $S^n_t$'s, the Stirling numbers of the second kind, we establish a formula for the sum of $(n-1)$ times partial sums of the $m$-th powers of an arithmetic progression in number theory.

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