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.


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 h-difference Riemann-Liouville operator mentioned in 9, 10 , the presence of the h parameter is particularly interesting from the numerical point of view, because when h 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 h sum of order m ≥ 1 Δ −m h f t 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 mth partial sums on nth powers of arithmetic, arithmetic-geometric progressions, and products of n consecutive terms of arithmetic progression using Δ −m u k 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

Preliminaries
Before stating and proving our results, we present some notations, basic definitions, and preliminary results which will be useful for further subsequent discussions.
where k/ denotes the integer part of k/ , N j {j, j, 2 j, . ..} and N 1 j N j .Throughout this paper, c j is a constant for all k ∈ N j and for any positive integer m, we denote , and so on.
Definition 2.1 see 13 .For a real valued function u k , the generalized difference operator Δ and its inverse are, respectively, defined as Definition 2.2 see 10 .For k, n ∈ 0, ∞ , the -factorial function is defined by Lemma 2.6.Let v k and w k 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 w k Δ z k in 2.9 .

Main Results
In this section, we use the following notations: Let n be any nonnegative integer.Then, In particular, when For any constant c, since 1 k 0 , by 3.2 and linearity of Δ −1 , Proof.The proof follows by taking 1, u k k n in 2.6 and 3.2 .
Theorem 3.5.Let m be a positive integer, ∈ 0, ∞ , and k ∈ m , ∞ .Then, Proof.Taking Δ −1 on 2.6 , and applying 2.6 for Δ −1 u k − r , we get From the notation given this section and ordering the terms u k − r , we find Again, taking Δ −1 on 3.8 , by 2.6 for Δ −1 u k − r , 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 m ∈ N 2 , ∈ 0, ∞ , and k ∈ m , ∞ .Then, 3.11 Proof.Applying the limit j to k on Δ −1 u k , we write where Δ −1 u j is constant and Δ −1 u k is a function of k.Taking Δ −1 on 3.12 , by 3.4 and applying the limit j to k, we obtain which can be expressed as

3.14
where t 1 and {m 1 m t 1} ∈ 1 L 1 and is same as

3.15
where all the terms except Δ −2 u k , k 2−m t are constants.

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Again taking Δ −1 on 3.15 , by 3.2 and 3.4 , we arrive which is the same as 3.17 where i 2. In the same way, we find which can be expressed as the following:

3.19
The proof completes by continuing this process.

3.20
Proof.The proof follows by equating 3.6 and 3.11 .

3.23
Taking Δ −1 on and applying 3.23 for m − 1 times, we arrive

3.24
The proof follows by taking u k k n e −k in Theorem 3.7.
The following example illustrates Corollary 3.8.
Example 3.9.Consider the case when m 4 and n 2. In this case,

3.25
The double summation expression of 3.25 will be obtained by adding the sums corresponds to 1 L 3 : 3.27 and to 3 L 3 :

3.29
Proof.Since k n e −k 0 as k → ∞ and 0 r 0 if r / 0 and 0 r 1 if r 0, the upper limit of 3.23 for k → ∞ will be zero and lower limit for j 0, gives 3.29 .

Discrete Gamma-Factorial Function
First we derive infinite series formula using Δ −1 , which induces the definition of discrete gamma factorial function.
Proof.From 2.6 , and expressing its terms in reverse order, we find which is the same as and the discrete k-gamma function is defined as

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In particular, when k 0, 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 ∈ 0, ∞ and n ∈ N.Then, Proof.The proof follows by taking j k, k ∞ in 3.29 and multiplying it by .
Theorem 4.4.Let ∈ 0, ∞ and n ∈ N 1 .Then, 1 − e − n 1 .4.9 Proof.From 2.7 , 3.1 , and 4.6 , we get Since k n e −k 0 for k 0 and k ∞, 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 Proof.The proof of 4.11 follows by 2.5 and linearity of Δ −1 .Now, 4.12 will be obtained by taking m 1 in 4.11 and using 4.6 , 4.7 , and 4.9 .

4.17
Now, the proof follows from 2.6 and 4.17 .Proof.From 2.1 and 2.2 , we find which yields by 2.2 ,

4.20
Now, the proof follows from 2.6 and 4.20 .

Theorem 4 . 5 .
Let S n r be as given in 2.5 and n ∈ N 1 .Then,Δ −m k n e −k