Advances in Numerical AnalysisVolume 2012 (2012), Article ID 780646, 13 pagesdoi:10.1155/2012/780646

Research Article

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

G. Britto Antony Xavier, V. Chandrasekar, S. U. Vasanthakumar, and B. Govindan

Department of Mathematics, Sacred Heart College, Tirupattur 635601, India

Received 17 July 2012; Accepted 5 October 2012

Academic Editor: Rüdiger Weiner

Copyright © 2012 G. Britto Antony Xavier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 difference become

Lemma 2.4 (see [13]). Let ’s be the Stirling numbers of second kind. Then,

Theorem 2.5 (see [13]). Let be real valued function. Then for ,

Lemma 2.6. Let and be two real valued functions. Then,

Proof. From (2.1), we find Applying (2.2) in (2.8), we obtain The proof follows by taking in (2.9).

3. Main Results

In this section, we use the following notations: = , is an empty set, , ,. In general, = set of all subsets of size from the set such that if , then , ,, power set of , for , and for .

Lemma 3.1 (see [13]). Let . Then,

Lemma 3.2 (see [13]). Let be any nonnegative integer. Then, In particular, when , , , then (3.2) becomes

Remark 3.3. For any constant , since , by (3.2) and linearity of ,

Lemma 3.4. Let and . Then,

Proof. The proof follows by taking , in (2.6) and (3.2).

Theorem 3.5. Let be a positive integer, , and . Then,

Proof. Taking on (2.6), and applying (2.6) for , we get From the notation given this section and ordering the terms , we find Again, taking on (3.8), by (2.6) for , we arrive which yields by (3.5), Now, (3.6) will be obtained by continuing this process and using (3.5).

Theorem 3.6. Let , , and . Then,

Proof. Applying the limit to on , we write where is constant and is a function of . Taking on (3.12), by (3.4) and applying the limit to , we obtain which can be expressed as where and and is same as where all the terms except , are constants. Again taking on (3.15), by (3.2) and (3.4), we arrive which is the same as where . In the same way, we find which can be expressed as the following: The proof completes by continuing this process.

Theorem 3.7 (partial summation formula). Let . Then,

Proof. The proof follows by equating (3.6) and (3.11).

Corollary 3.8. Let , , and . Then,

Proof. Taking , , in (2.7), by (2.4), (3.1), and (3.2), we get By (3.1) and applying (2.7) for , , we arrive Taking on and applying (3.23) for times, we arrive The proof follows by taking in Theorem 3.7.

The following example illustrates Corollary 3.8.

Example 3.9. Consider the case when and . In this case, , , , and (3.21) becomes The double summation expression of (3.25) will be obtained by adding the sums corresponds to : corresponds to : and to :

Corollary 3.10. Let and . Then,

Proof. Since as and if and if , the upper limit of (3.23) for will be zero and lower limit for , gives (3.29).

4. Discrete Gamma—Factorial Function

First we derive infinite series formula using , which induces the definition of discrete gamma factorial function.

Theorem 4.1. Let and . Then,

Proof. From (2.6), and expressing its terms in reverse order, we find which is the same as Now (4.1) follows by given condition and then replacing by .

Definition 4.2. For and , the discrete -gamma factorial function is defined as and the discrete -gamma function is defined as In particular, when , (4.4) and (4.5) becomes which can be called as the discrete gamma factorial function and the discrete gamma function, respectively.

Theorem 4.3. Let and . Then,

Proof. The proof follows by taking , in (3.29) and multiplying it by .

Theorem 4.4. Let and . Then,

Proof. From (2.7), (3.1), and (4.6), we get Since for and , (4.10) gives first part of (4.9) by (4.6). Now second part of (4.9) will be obtained by applying first part of (4.9) again and again and using the identity .

Theorem 4.5. Let be as given in (2.5) and . Then, which gives

Proof. The proof of (4.11) follows by (2.5) and linearity of . Now, (4.12) will be obtained by taking in (4.11) and using (4.6), (4.7), and (4.9).

Theorem 4.6. Let and . Then,

Proof. As , all the terms except the last term of (4.12) will be zero. Also, since , by (4.9), we arrive Now, (4.13) follows by replacing by and taking limit on (4.9).

Theorem 4.7. Let and . Then,

Proof. From (2.1) and (2.2), we have which yields Now, the proof follows from (2.6) and (4.17).

Theorem 4.8. Let and . Then,

Proof. From (2.1) and (2.2), we find which yields by (2.2), Now, the proof follows from (2.6) and (4.20).

Corollary 4.9. Let and. Then,

Proof. The proof follows from (4.1) and (4.20).

Corollary 4.10. (i) When , (ii) When and , (iii) When , ,

Acknowledgment

This paper is supported by the University Grants Commission, SERO, Hyderabad, India.

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