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

Abstract

The recent theory and applications of difference operator Δ_𝐿 introduced in (M. Maria Susai Manuel et al., 2012) are enriched and extended with a useful tool Δ_ℓ^−𝑚[𝑘_ℓ^(𝑛)𝑒^−𝑘]||^𝑘_{(𝑚−1)ℓ+𝑗} 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 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_ℎ^−𝑚𝑓)(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_ℓ^−𝑚𝑢(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 and gamma factorial function in number theory by getting closed and summation form of Δ_ℓ^−𝑚𝑢(𝑘)‖^𝑘_{(𝑚−1)ℓ+𝑗}, (here we replace 𝑓(𝑡) by 𝑢(𝑘), ℎ by ℓ, and 𝜈 by 𝑚 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 ℓ>0,𝑘∈[0,∞),𝑗=𝑘−[𝑘/ℓ]ℓ where [𝑘/ℓ] denotes the integer part of 𝑘/ℓ,ℕ_ℓ(𝑗)={𝑗,ℓ+𝑗,2ℓ+𝑗,…} and ℕ₁(𝑗)=ℕ(𝑗). Throughout this paper, 𝑐_𝑗 is a constant for all 𝑘∈ℕ_ℓ(𝑗) and for any positive integer 𝑚, we denote Δ_ℓ^−𝑚𝑢(𝑘)‖^𝑘_{(𝑚−1)ℓ+𝑗} = Δ_ℓ⁻¹(⋯Δ_ℓ⁻¹(Δ_ℓ⁻¹𝑢(𝑘)|^𝑘_𝑗)|^𝑘_ℓ+𝑗⋯)|^𝑘_{(𝑚−1)ℓ+𝑗}, where Δ_ℓ⁻¹𝑢(𝑘)|^𝑘_𝑗 = 𝑢₁(𝑘) = Δ_ℓ⁻¹𝑢(𝑘)−Δ_ℓ⁻¹𝑢(𝑗),Δ_ℓ⁻¹(Δ_ℓ⁻¹𝑢(𝑘)|^𝑘_𝑗)|^𝑘_ℓ+𝑗 = 𝑢₂(𝑘) = Δ_ℓ⁻¹𝑢₁(𝑘)−Δ_ℓ⁻¹𝑢₁(ℓ+𝑗), and so on.

Definition 2.1 (see [13]). For a real valued function 𝑢(𝑘), the generalized difference operator Δ_ℓ and its inverse are, respectively, defined as Δ_ℓ𝑢(𝑘)=𝑢(𝑘+ℓ)−𝑢(𝑘),𝑘∈[0,∞),ℓ∈(0,∞),ifΔ_ℓ𝑣(𝑘)=𝑢(𝑘),then𝑣(𝑘)=Δ_ℓ⁻¹𝑢(𝑘)+𝑐_𝑗. (2.1)(2.2)

Definition 2.2 (see [10]). For 𝑘,𝑛∈(0,∞), the ℓ-factorial function is defined by 𝑘_ℓ^(𝑛)=ℓ^𝑛Γ(𝑘/ℓ+1)Γ(),𝑘/ℓ+1−𝑛 (2.3) where Γ is the Euler gamma function and 𝑘₁^(𝑛)=𝑘^(𝑛).

Remark 2.3. When 𝑛∈ℕ(1), (2.3), and its Δ_ℓ difference become 𝑘_ℓ^(𝑛)=^𝑛−1_𝑡=0(𝑘−𝑡ℓ),Δ_ℓ𝑘_ℓ^(𝑛)=(𝑛ℓ)𝑘_ℓ^(𝑛−1). (2.4)

Lemma 2.4 (see [13]). Let 𝑆^𝑛_𝑟’s be the Stirling numbers of second kind. Then, 𝑘^𝑛=^𝑛_𝑟=1𝑆^𝑛_𝑟ℓ^𝑛−𝑟𝑘_ℓ^(𝑟). (2.5)

Theorem 2.5 (see [13]). Let 𝑢(𝑘),𝑘∈[0,∞) be real valued function. Then for 𝑘∈[ℓ,∞), Δ_ℓ⁻¹𝑢(𝑘)^𝑘_𝑗=^[𝑘/ℓ]_𝑟=1𝑢(𝑘−𝑟ℓ). (2.6)

Lemma 2.6. Let 𝑣(𝑘) and 𝑤(𝑘) be two real valued functions. Then, Δ_ℓ⁻¹[𝑣(𝑘)𝑤(𝑘)]=𝑣(𝑘)Δ_ℓ⁻¹𝑤(𝑘)−Δ_ℓ⁻¹Δ_ℓ⁻¹𝑤(𝑘+ℓ)Δ_ℓ𝑣(𝑘). (2.7)

Proof. From (2.1), we find Δ_ℓ[𝑣(𝑘)𝑧(𝑘)]=𝑧(𝑘+ℓ)Δ_ℓ𝑣(𝑘)+𝑣(𝑘)Δ_ℓ𝑧(𝑘). (2.8) Applying (2.2) in (2.8), we obtain Δ_ℓ⁻¹𝑣(𝑘)Δ_ℓ𝑧(𝑘)=𝑣(𝑘)𝑧(𝑘)−Δ_ℓ⁻¹𝑧(𝑘+ℓ)Δ_ℓ𝑣(𝑘). (2.9) The proof follows by taking 𝑤(𝑘)=Δ_ℓ𝑧(𝑘) in (2.9).

3. Main Results

In this section, we use the following notations: 𝐿_𝑚−1={1,2,…,𝑚−1},0(𝐿_𝑚−1) = {𝜙}, 𝜙 is an empty set, 1(𝐿_𝑚−1)={{1},{2},…,{𝑚−1}}, 2(𝐿_𝑚−1)={{1,2},{1,3},…,{1,𝑚−1},{2,3},…,{2,𝑚−1},…,{𝑚−2,𝑚−1}}. In general, 𝑡(𝐿_𝑚−1) = set of all subsets of size 𝑡 from the set 𝐿_𝑚−1 such that if {𝑚₁,𝑚₂,…,𝑚_𝑡}∈𝑡(𝐿_𝑚−1), then 𝑚₁<𝑚₂<⋯<𝑚_𝑡, (𝑚−1)(𝐿_𝑚−1)={{1,2,…,𝑚−1}},℘(𝐿_𝑚−1)=⋃^𝑚−1_𝑡=0𝑡(𝐿_𝑚−1), power set of 𝐿_𝑚−1, ∑^𝑚−1_𝑡=1𝑓(𝑡)=0 for 𝑚≤1, and ∏^𝑡_𝑖=2𝑓(𝑖)=1 for 𝑡≤1.

Lemma 3.1 (see [13]). Let 𝑘∈[ℓ,∞). Then, Δ_ℓ⁻¹𝑒^−𝑘𝑘_𝑗=𝑒^−𝑘𝑒^−ℓ−𝑒−1𝑒^{−𝑗−ℓ}.−1 (3.1)

Lemma 3.2 (see [13]). Let 𝑛 be any nonnegative integer. Then, Δ_ℓ⁻¹𝑘_ℓ^(𝑛)𝑘_𝑗=𝑘_ℓ^(𝑛+1)ℓ()−𝑗𝑛+1ℓ(^ℓ(𝑛+1)).𝑛+1 (3.2) In particular, when ℓ=1, 𝑘=(𝑟−1)∈ℕ={1,2,3,…}, 𝑗=0, then (3.2) becomes Δ⁻¹𝑘^(𝑛)₀^𝑟−1=(𝑟−1)^(𝑛+1)().𝑛+1 (3.3)

Remark 3.3. For any constant 𝑐, since 1=𝑘_ℓ⁽⁰⁾, by (3.2) and linearity of Δ_ℓ⁻¹, Δ_ℓ⁻¹𝑐^𝑘_𝑗=𝑐Δ_ℓ⁻¹𝑘_ℓ^(0)𝑘_𝑗𝑘=𝑐^ℓ(1)−𝑗1!ℓ^ℓ(1).1!ℓ (3.4)

Lemma 3.4. Let 𝑛∈ℕ(0) and 𝑟∈ℕ(3). Then, Δ⁻¹𝑘^(𝑛)₀^𝑟−1=(𝑟−2)^(𝑛)+(𝑟−3)^(𝑛)+⋯+(1)^(𝑛)=(𝑟−1)^(𝑛+1)().𝑛+1 (3.5)

Proof. The proof follows by taking ℓ=1, 𝑢(𝑘)=𝑘^(𝑛) in (2.6) and (3.2).

Theorem 3.5. Let 𝑚 be a positive integer, ℓ∈(0,∞), and 𝑘∈[𝑚ℓ,∞). Then, Δ_ℓ^−𝑚𝑢(𝑘)^𝑘_{()𝑚−1ℓ+𝑗}=^[𝑘/ℓ]_𝑟=𝑚(𝑟−1)^(𝑚−1)()!𝑢().𝑚−1𝑘−𝑟ℓ (3.6)

Proof. Taking Δ_ℓ⁻¹ on (2.6), and applying (2.6) for Δ_ℓ⁻¹𝑢(𝑘−𝑟ℓ), we get Δ_ℓ⁻¹Δ_ℓ⁻¹𝑢(𝑘)^𝑘_𝑗^𝑘_ℓ+𝑗=^[𝑘/ℓ]_𝑟=1Δ_ℓ⁻¹𝑢(𝑘−𝑟ℓ)=^[𝑘/ℓ]_𝑟=1^{[(𝑘−𝑟ℓ)/ℓ]}_𝑠=1𝑢(𝑘−𝑟ℓ−𝑠ℓ). (3.7) From the notation given this section and ordering the terms 𝑢(𝑘−𝑟ℓ), we find Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗=^[𝑘/ℓ]_𝑟=2(𝑟−1)⁽¹⁾𝑢().1!𝑘−𝑟ℓ (3.8) Again, taking Δ_ℓ⁻¹ on (3.8), by (2.6) for Δ_ℓ⁻¹𝑢(𝑘−𝑟ℓ), we arrive Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗=^[𝑘/ℓ]_𝑟=3(𝑟−2)⁽¹⁾+()1!𝑟−3⁽¹⁾(1)1!+⋯+⁽¹⁾𝑢(),1!𝑘−𝑟ℓ (3.9) which yields by (3.5), Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗=^[𝑘/ℓ]_𝑟=3(𝑟−1)⁽²⁾𝑢().2!𝑘−𝑟ℓ (3.10) Now, (3.6) will be obtained by continuing this process and using (3.5).

Theorem 3.6. Let 𝑚∈ℕ(2), ℓ∈(0,∞), and 𝑘∈[𝑚ℓ,∞). Then, Δ_ℓ^−𝑚𝑢(𝑘)^𝑘_{()𝑚−1ℓ+𝑗}=Δ_ℓ^−𝑚𝑢(𝑘)^𝑘_{()𝑚−1ℓ+𝑗}+^𝑚−1_{𝑡=1𝑚1,…,𝑚𝑡𝐿∈𝑡𝑚−1}(−1)^𝑡×Δ^−𝑚1_ℓ𝑢𝑚₁𝑘−1ℓ+𝑗^{(𝑚−𝑚𝑡)ℓ}𝑚−𝑚_𝑡!ℓ^{𝑚−𝑚𝑡}×^𝑡_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_{()𝑚−1ℓ+𝑗}. (3.11)

Proof. Applying the limit 𝑗 to 𝑘 on Δ_ℓ⁻¹𝑢(𝑘), we write Δ_ℓ⁻¹𝑢(𝑘)^𝑘_𝑗=Δ_ℓ⁻¹𝑢(𝑘)−Δ_ℓ⁻¹𝑢𝑗, (3.12) where Δ_ℓ⁻¹𝑢(𝑗) is constant and Δ_ℓ⁻¹𝑢(𝑘) is a function of 𝑘. Taking Δ_ℓ⁻¹ on (3.12), by (3.4) and applying the limit ℓ+𝑗 to 𝑘, we obtain Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗=Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗−Δ_ℓ⁻¹𝑢𝑗𝑘_ℓ⁽¹⁾1!ℓ^𝑘_ℓ+𝑗, (3.13) which can be expressed as Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗=Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗+(−1)^𝑡Δ^−𝑚1_ℓ𝑢𝑚₁𝑘−1ℓ+𝑗^{(2−𝑚𝑡)ℓ}2−𝑚_𝑡!ℓ^{2−𝑚𝑡𝑘}_ℓ+𝑗, (3.14) where 𝑡=1 and {𝑚₁=𝑚_𝑡=1}∈1(𝐿₁) and is same as Δ_ℓ⁻²𝑢(𝑘)^𝑘_ℓ+𝑗=Δ_ℓ⁻²𝑢(𝑘)−Δ_ℓ⁻²𝑢+()ℓ+𝑗−1^𝑡𝑘^{(2−𝑚𝑡)}_ℓ2−𝑚_𝑡!ℓ^2−𝑚𝑡Δ^−𝑚1_ℓ𝑢𝑚₁−()−1ℓ+𝑗−1^𝑡ℓ+𝑗^{(2−𝑚𝑡)}_ℓ2−𝑚_𝑡!ℓ^2−𝑚𝑡Δ^−𝑚1_ℓ𝑢𝑚₁−1ℓ+𝑗, (3.15) where all the terms except Δ_ℓ⁻²𝑢(𝑘), 𝑘^{(2−𝑚𝑡)}_ℓ are constants.
Again taking Δ_ℓ⁻¹ on (3.15), by (3.2) and (3.4), we arrive Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗=Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗−Δ_ℓ⁻²𝑢𝑘ℓ+𝑗^ℓ(1)1!ℓ^𝑘_2ℓ+𝑗+(−1)¹Δ^−𝑚1_ℓ𝑢𝑚₁𝑘−1ℓ+𝑗^{(3−𝑚𝑡)ℓ}3−𝑚_𝑡!ℓ^{3−𝑚𝑡𝑘}_2ℓ+𝑗+(−1)²Δ^−𝑚1_ℓ𝑢𝑚₁𝑘−1ℓ+𝑗^ℓ(1)1!ℓℓ+𝑗^{(2−𝑚𝑡)}_ℓ2−𝑚_𝑡!ℓ^{2−𝑚𝑡𝑘}_2ℓ+𝑗, (3.16) which is the same as Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗=Δ_ℓ⁻³𝑢(𝑘)^𝑘_2ℓ+𝑗+²_{𝑡=1𝑚𝑡,𝑚=1𝑡𝐿=2∈12𝑚1=1,𝑚𝑡𝐿=2∈22}(−1)^𝑡Δ^−𝑚1_ℓ𝑢𝑚₁×𝑘−1ℓ+𝑗_{(3−𝑚𝑡)ℓ}3−𝑚_𝑡!ℓ^3−𝑚𝑡𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_2ℓ+𝑗, (3.17) where 𝑖=2. In the same way, we find Δ_ℓ⁻⁴𝑢(𝑘)^𝑘_3ℓ+𝑗=Δ_ℓ⁻⁴𝑢(𝑘)^𝑘_3ℓ+𝑗+³_{𝑡=1𝑚𝑡𝐿∈13,𝑚1,𝑚𝑡𝐿∈23𝑚1,𝑚2,𝑚𝑡𝐿∈33}(−1)^𝑡Δ^−𝑚1_ℓ𝑢𝑚₁×𝑘−1ℓ+𝑗_{(4−𝑚𝑡)ℓ}4−𝑚_𝑡!ℓ^{4−𝑚𝑡3}_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_3ℓ+𝑗, (3.18) which can be expressed as the following:Δ_ℓ⁻⁴𝑢(𝑘)^𝑘_3ℓ+𝑗=Δ_ℓ⁻⁴𝑢(𝑘)^𝑘_3ℓ+𝑗+³_{𝑡=1𝑚1,…,𝑚𝑡𝐿∈𝑡3}(−1)^𝑡Δ^−𝑚1_ℓ𝑢𝑚₁×𝑘−1ℓ+𝑗_{(4−𝑚𝑡)ℓ}4−𝑚_𝑡!ℓ^{4−𝑚𝑡3}_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_3ℓ+𝑗. (3.19) The proof completes by continuing this process.

Theorem 3.7 (partial summation formula). Let 𝑚∈ℕ(2). Then, ^[𝑘/ℓ]_𝑟=𝑚(𝑟−1)^(𝑚−1)()!𝑢()𝑚−1𝑘−𝑟ℓ=Δ_ℓ^−𝑚𝑢(𝑘)^𝑘_{()𝑚−1ℓ+𝑗}+^𝑚−1_{𝑡=1𝑚1,…,𝑚𝑡𝐿∈𝑡𝑚−1}(−1)^𝑡×Δ^−𝑚1_ℓ𝑢𝑚₁𝑘−1ℓ+𝑗^{(𝑚−𝑚𝑡)ℓ}𝑚−𝑚_𝑡!ℓ^{𝑚−𝑚𝑡}×^𝑡_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_{()𝑚−1ℓ+𝑗}. (3.20)

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

Corollary 3.8. Let 𝑚∈ℕ(2), ℓ∈(0,∞), and 𝑘∈[𝑚ℓ,∞). Then, ^[𝑘/ℓ]_𝑟=𝑚(𝑟−1)^(𝑚−1)(𝑚−1)!𝑒^{(𝑘−𝑟ℓ)}(𝑘−𝑟ℓ)_ℓ^(𝑛)=^𝑛+1_𝑟=1(−1)^𝑟−1𝑛^(𝑟−1)()!()𝑟−1𝑚+𝑟−2^(𝑟−1)ℓ^𝑟−1𝑒^−ℓ−1^{𝑟+𝑚−1}×𝑘_ℓ^{(𝑛+1−𝑟)}𝑒^{(𝑘+(𝑟−1)ℓ)}+^𝑚−1_{𝑡=1𝑚1,…,𝑚𝑡𝐿∈𝑡𝑚−1}×(−1)^𝑡𝑚₁+𝑟−2^(𝑟−1)ℓ^𝑟−1𝑒^−ℓ−1^{𝑟+𝑚1−1}×𝑚₁−1ℓ+𝑗_ℓ^{(𝑛+1−𝑟)}𝑘^{(𝑚−𝑚𝑡)}_ℓ𝑚−𝑚_𝑡!ℓ^{𝑚−𝑚𝑡}𝑒^{((𝑚1−1)ℓ+𝑗+(𝑟−1)ℓ)}×^𝑡_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_{()𝑚−1ℓ+𝑗}. (3.21)

Proof. Taking 𝑣(𝑘)=𝑘_ℓ^(𝑛), 𝑤(𝑘)=𝑒^−𝑘, 𝑢(𝑘)=𝑣(𝑘)𝑤(𝑘) in (2.7), by (2.4), (3.1), and (3.2), we get Δ_ℓ⁻¹𝑢(𝑘)=𝑘_ℓ^(𝑛)𝑒^−𝑘𝑒^−ℓ−1−Δ_ℓ⁻¹𝑒^−𝑘+ℓ𝑒^−ℓ()𝑘−1𝑛ℓ_ℓ^(𝑛−1). (3.22) By (3.1) and applying (2.7) for 𝑘_ℓ^(𝑛−1)𝑒^−𝑘, 𝑘_ℓ^(𝑛−2)𝑒^−𝑘,…,𝑘_ℓ⁽¹⁾𝑒^−𝑘, we arrive Δ_ℓ⁻¹𝑢(𝑘)=^𝑛+1_𝑟=1(−1)^𝑟−1𝑛^(𝑟−1)()!()𝑟−1𝑟−1^(𝑟−1)ℓ^𝑟−1𝑒^−ℓ−1^𝑟𝑘_ℓ^{(𝑛+1−𝑟)}𝑒^{(𝑘+(𝑟−1)ℓ)}. (3.23) Taking Δ_ℓ⁻¹ on and applying (3.23) for (𝑚−1) times, we arrive Δ_ℓ^−𝑚𝑢(𝑘)=^𝑛+1_𝑟=1(−1)^𝑟−1𝑛^(𝑟−1)()!()𝑟−1𝑚+𝑟−2^(𝑟−1)ℓ^𝑟−1𝑒^−ℓ−1^{𝑟+𝑚−1}𝑘_ℓ^{(𝑛+1−𝑟)}𝑒^{(𝑘+(𝑟−1)ℓ)}. (3.24) The proof follows by taking 𝑢(𝑘)=𝑘_ℓ^(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, 𝐿₃={1,2,3}, 1(𝐿₃)={{1},{2},{3}}, 2(𝐿₃)={{1,2},{1,3},{2,3}}, 3(𝐿₃)={{1,2,3}} and (3.21) becomes ^[𝑘/ℓ]_𝑟=4(𝑟−1)⁽³⁾(𝑘−𝑟ℓ)_ℓ⁽²⁾(3)!𝑒^{(𝑘−𝑟ℓ)}=³_𝑟=1(−1)^𝑟−12^(𝑟−1)()!()𝑟−12+𝑟^(𝑟−1)ℓ^𝑟−1𝑘_ℓ^(1+𝑟)𝑒^−ℓ−1^𝑟+3𝑒^{(𝑘+(𝑟−1)ℓ)}+³_{𝑡=1𝑚1,…,𝑚𝑡𝐿∈𝑡3}(−1)^𝑡𝑚₁+𝑟−2^(𝑟−1)ℓ^𝑟−1𝑒^−ℓ−1^{𝑟+𝑚1−1}×𝑚₁−1ℓ+𝑗_ℓ^(1+𝑟)𝑘^{(4−𝑚𝑡)}_ℓ4−𝑚_𝑡!ℓ^4−𝑚𝑡𝑒^{((𝑚1𝑡−1)ℓ+𝑗+(𝑟−1)ℓ)}_𝑖=2𝑚_𝑖−1ℓ+𝑗^{(𝑚𝑖−𝑚𝑖−1)}_ℓ𝑚_𝑖−𝑚_𝑖−1!ℓ^{𝑚𝑖−𝑚𝑖−1𝑘}_3ℓ+𝑗. (3.25) The double summation expression of (3.25) will be obtained by adding the sums corresponds to 1(𝐿₃): (𝑟−1)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟}𝑒^{(𝑗+(𝑟−1)ℓ)}𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽³⁾3!ℓ³+(𝑟)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟+1}𝑒^{(ℓ+𝑗+(𝑟−1)ℓ)}ℓ+𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽²⁾2!ℓ²+(𝑟+1)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟+2}𝑒^{(2ℓ+𝑗+(𝑟−1)ℓ)}2ℓ+𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽¹⁾ℓ, (3.26) corresponds to 2(𝐿₃): (𝑟−1)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟}𝑒^{(𝑗+(𝑟−1)ℓ)}𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽²⁾2!ℓ²ℓ+𝑗_ℓ⁽¹⁾ℓ+(𝑟−1)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟}e^{(𝑗+(𝑟−1)ℓ)}𝑗_ℓ^(1+𝑟)×𝑘_ℓ⁽¹⁾ℓ2ℓ+𝑗_ℓ⁽²⁾2!ℓ²+(𝑟)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟+1}𝑒^{(ℓ+𝑗+(𝑟−1)ℓ)}ℓ+𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽¹⁾ℓ2ℓ+𝑗_ℓ⁽²⁾2!ℓ², (3.27) and to 3(𝐿₃): (𝑟−1)^(𝑟−1)ℓ^𝑟−1𝑒^{−ℓ−1𝑟}𝑒^{(𝑗+(𝑟−1)ℓ)}𝑗_ℓ^(1+𝑟)𝑘_ℓ⁽¹⁾ℓℓ+𝑗_ℓ⁽¹⁾ℓ2ℓ+𝑗_ℓ⁽¹⁾ℓ. (3.28)

Corollary 3.10. Let ℓ∈(0,∞) and 𝑘∈[ℓ,∞). Then, Δ_ℓ⁻¹𝑘_ℓ^(𝑛)𝑒^−𝑘𝑘_𝑗=^[𝑘/ℓ]_𝑟=1(𝑘−𝑟ℓ)_ℓ^(𝑛)𝑒^{(𝑘−𝑟ℓ)}=^𝑛+1_𝑟=1−𝑛^(𝑟−1)ℓ^𝑟−1𝑘_ℓ^{(𝑛+1−𝑟)}1−𝑒^−ℓ𝑟𝑒^{(𝑘+(𝑟−1)ℓ)𝑘}_𝑗. (3.29)

Proof. Since 𝑘_ℓ^(𝑛)𝑒^−𝑘=0 as 𝑘→∞ and 0_ℓ^(𝑟)=0 if 𝑟≠0 and 0_ℓ^(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).

4. Discrete Gamma—Factorial Function

First we derive infinite series formula using Δ_ℓ⁻¹, which induces the definition of discrete gamma factorial function.

Theorem 4.1. Let 𝑘∈[0,∞) and lim_𝑘→∞Δ_ℓ⁻¹𝑢(𝑘)=0. Then, Δ_ℓ⁻¹𝑢(𝑘)^∞_𝑘=^∞_𝑟=0𝑢(𝑘+𝑟ℓ). (4.1)

Proof. From (2.6), and expressing its terms in reverse order, we find Δ_ℓ⁻¹𝑢(𝑘)^𝑘_𝑗+Δ_ℓ⁻¹𝑢(𝑘)^∞_𝑘=𝑢𝑗+𝑢ℓ+𝑗+⋯+𝑢(𝑘−ℓ)+𝑢(𝑘)+𝑢(𝑘+ℓ)+⋯+𝑢(∞), (4.2) which is the same as Δ_ℓ⁻¹𝑢(𝑘)^∞_𝑗=^∞_𝑟=0𝑢𝑗+𝑟ℓ. (4.3) Now (4.1) follows by given condition Δ_ℓ⁻¹𝑢(∞)=0 and then replacing 𝑗 by 𝑘.

Definition 4.2. For 𝑛∈(−∞,∞)−{0} and 𝑘∈[0,∞), the discrete 𝑘-gamma factorial function is defined as _𝑘Γ_ℓ((𝑛))=ℓΔ_ℓ⁻¹𝑘_ℓ^(𝑛−1)𝑒^−𝑘∞_𝑘=ℓ^∞_𝑟=0(𝑘+𝑟ℓ)_ℓ^(𝑛−1)𝑒^{−(𝑘+𝑟ℓ)}, (4.4) and the discrete 𝑘-gamma function is defined as _𝑘Γ_ℓ(𝑛)=ℓΔ_ℓ⁻¹𝑘^𝑛−1𝑒^−𝑘∞_𝑘=ℓ^∞_𝑟=0(𝑘+𝑟ℓ)^𝑛−1𝑒^{−(𝑘+𝑟ℓ)}. (4.5) In particular, when 𝑘=0, (4.4) and (4.5) becomes Γ_ℓ((𝑛))=ℓΔ_ℓ⁻¹𝑘_ℓ^(𝑛−1)𝑒^−𝑘∞_𝑘=0=ℓ^∞_𝑟=1(𝑟ℓ)_ℓ^(𝑛−1)𝑒^−𝑟ℓ,Γ_ℓ(𝑛)=ℓΔ_ℓ⁻¹𝑘^𝑛−1𝑒^−𝑘∞_𝑘=0=ℓ^∞_𝑟=1(𝑟ℓ)^𝑛−1𝑒^−𝑟ℓ (4.6)(4.7) which can be called as the discrete gamma factorial function and the discrete gamma function, respectively.

Theorem 4.3. Let ℓ∈(0,∞) and 𝑛∈ℕ. Then, _𝑘Γ_ℓ((𝑛+1))=ℓ^𝑛_𝑟=0(𝑛)^(𝑟)ℓ^𝑟𝑘_ℓ^{(𝑛−𝑟)}𝑒^{−(𝑘+𝑟ℓ)}1−𝑒^−ℓ𝑟+1=ℓ^∞_𝑟=0(𝑘+𝑟ℓ)_ℓ^(𝑛)𝑒^{−(𝑘+𝑟ℓ)}. (4.8)

Proof. The proof follows by taking 𝑗=𝑘, 𝑘=∞ in (3.29) and multiplying it by ℓ.

Theorem 4.4. Let ℓ∈(0,∞) and 𝑛∈ℕ(1). Then, Γ_ℓ((𝑛+1))=ℓ𝑒^−ℓ1−𝑒^−ℓ𝑛Γ_ℓ𝑛ℓ(())=𝑛!^𝑛+1𝑒^−𝑛ℓ1−𝑒^−ℓ𝑛+1. (4.9)

Proof. From (2.7), (3.1), and (4.6), we get Γ_ℓ((𝑛+1))=ℓ𝑘_ℓ^(𝑛)𝑒^−𝑘𝑒^−ℓ−1−Δ_ℓ⁻¹𝑒^{−(𝑘+ℓ)}𝑒^−ℓ()𝑘−1𝑛ℓ_ℓ^{(𝑛−1)∞}_𝑘=0. (4.10) Since 𝑘_ℓ^(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 Γ_ℓ((1))=ℓ/(1−𝑒^−ℓ).

Theorem 4.5. Let 𝑆^𝑛_𝑟 be as given in (2.5) and 𝑛∈ℕ(1). Then, Δ_ℓ^−𝑚𝑘^𝑛𝑒^−𝑘=^𝑛_𝑟=1𝑆^𝑛_𝑟ℓ^𝑛−𝑟Δ_ℓ^−𝑚𝑘_ℓ^(𝑟)𝑒^−𝑘, (4.11) which gives Γ_ℓ(𝑛+1)=^𝑛_𝑟=1𝑆^𝑛_𝑟ℓ^𝑛−𝑟Γ_ℓ((𝑟+1))=ℓ𝑒^−ℓ1−𝑒^𝑛−ℓ_𝑟=1𝑆^𝑛_𝑟ℓ^𝑛−𝑟𝑟Γ_ℓ((𝑟)). (4.12)

Proof. The proof of (4.11) follows by (2.5) and linearity of Δ_ℓ⁻¹. Now, (4.12) will be obtained by taking 𝑚=1 in (4.11) and using (4.6), (4.7), and (4.9).

Theorem 4.6. Let ℓ∈(0,∞) and 𝑛∈ℕ(1). Then, lim_ℓ→0Γ_ℓ(𝑛)=lim_ℓ→0Γ_ℓ((𝑛))=(𝑛−1)!=Γ(𝑛). (4.13)

Proof. As ℓ→0, all the terms except the last term of (4.12) will be zero. Also, since lim_𝑘→∞ℓ𝑒^−ℓ/(1−𝑒^−ℓ)=1, by (4.9), we arrive lim_ℓ→0Γ_ℓ(𝑛)=lim_ℓ→0(𝑛−1)Γ_ℓ((𝑛+1))=lim_ℓ→0Γ_ℓ((𝑛)). (4.14) Now, (4.13) follows by replacing 𝑛 by 𝑛−1 and taking limit ℓ→0 on (4.9).

Theorem 4.7. Let 𝑘∈[ℓ,∞) and 𝑗=𝑘−[𝑘/ℓ]ℓ. Then, ^[𝑘/ℓ]_𝑟=1𝑘ℓΓ𝑘−𝑟ℓ𝑘−𝑟+1=Γℓ𝑗+1−Γℓ+1. (4.15)

Proof. From (2.1) and (2.2), we have Δ_ℓΓ𝑘ℓ=𝑘+1ℓΓ𝑘ℓ+1, (4.16) which yields Δ_ℓ⁻¹𝑘ℓΓ𝑘ℓ+1^𝑘_𝑗𝑘=Γℓ+1. (4.17) Now, the proof follows from (2.6) and (4.17).

Theorem 4.8. Let 𝑘∈[ℓ,∞) and 𝑗=𝑘−[𝑘/ℓ]ℓ. Then, ^[𝑘/ℓ]_𝑟=1(𝑘/ℓ)−𝑟()1𝑘/ℓ−r+1Γ))=1((𝑘/ℓ−𝑟+1Γ−1𝑗/ℓ+1Γ)).((𝑘/ℓ+1 (4.18)

Proof. From (2.1) and (2.2), we find Δ_ℓ1Γ))()((𝑘/ℓ+1=−𝑘/ℓ))Γ)),((𝑘/ℓ+1((𝑘/ℓ+1 (4.19) which yields by (2.2), −Δ_ℓ⁻¹(𝑘/ℓ)((𝑘/ℓ)+1)Γ((𝑘/ℓ)+1)^𝑘_𝑗=1Γ))−1((𝑘/ℓ+1Γ.𝑗/ℓ+1 (4.20) Now, the proof follows from (2.6) and (4.20).

Corollary 4.9. Let 𝑘∈[ℓ,∞)and𝑗=𝑘−[𝑘/ℓ]ℓ. Then, ^∞_𝑟=0(𝑘/ℓ)+𝑟()1𝑘/ℓ+𝑟+1Γ))=1((𝑘/ℓ+𝑟+1Γ)).((𝑘/ℓ+1 (4.21)

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

Corollary 4.10. (i) When 𝑘=0, ^∞_𝑟=0𝑟1𝑟+1Γ(𝑟+1)=1. (4.22)
(ii) When 𝑘=1 and ℓ=0.5, ^∞_𝑟=02+𝑟13+𝑟Γ()=13+𝑟=1Γ3.2! (4.23)
(iii) When 𝑘=𝑛ℓ, 𝑛∈𝑁(1), ^∞_𝑟=0𝑛+𝑟1𝑛+1+𝑟Γ()=1𝑛+1+𝑟Γ()=1𝑛+1.𝑛! (4.24)

Acknowledgment

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