Discrete Gamma (Factorial) Function and Its Series in Terms of a Generalized Difference Operator

The recent theory and applications of difference operator introduced in (M. Maria Susai Manuel et al., 2012) are enriched and extended with a useful tool for finding the values of various series of discrete gamma functions in number theory. Illustrative examples show the effectiveness of the obtained results in finding the values of various gamma series. 1. Introduction The fractional calculus involving gamma function is a generalization of differential calculus, allowing to define derivatives of real or complex order [1, 2]. It is a mathematical subject that has proved to be very useful in applied fields such as economics, engineering, and physics [3–7]. In 1989, Miller and Ross introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional difference operator [8]. In the general fractional -difference Riemann-Liouville operator mentioned in [9, 10], the presence of the parameter is particularly interesting from the numerical point of view, because when tends to zero the solutions of the fractional difference equations can be seen as approximations to the solutions of corresponding Riemann-Liouville fractional differential equation [9, 11]. On the other hand, fractional sum of order ( Definition 2.8 of [9]) is very useful to derive many interesting results in a different way in number theory such as the sum of the th partial sums on th powers of arithmetic, arithmetic-geometric progressions, and products of consecutive terms of arithmetic progression using [12]. We observed that no results in number theory using definition 2.8 of [9] had been derived. In this paper, we use Definition 2.8 of [9] in a different way and define discrete gamma factorial function to obtain summation formulas of certain series on gamma function and gamma factorial function in number theory by getting closed and summation form of , (here we replace by , by , and by on the notations used in [9]). 2. Preliminaries Before stating and proving our results, we present some notations, basic definitions, and preliminary results which will be useful for further subsequent discussions. Let where denotes the integer part of and . Throughout this paper, is a constant for all and for any positive integer , we denote = , where = = = = , and so on. Definition 2.1 (see [13]). For a real valued function , the generalized difference operator and its inverse are, respectively, defined as Definition 2.2 (see [10]). For , the -factorial function is defined by where is the Euler gamma function and . Remark 2.3. When , (2.3), and its