Some Properties of the 𝑞-Extension of the 𝑝-Adic Gamma Function

Abstract

We study the 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞. We give a new identity for the 𝑞-extension of the 𝑝-adic gamma Γ_𝑝,𝑞 in the case 𝑝=2. Also, we derive some properties and new representations of the 𝑞-extension of the 𝑝-adic gamma Γ_𝑝,𝑞 in general case.

1. Introduction

Let 𝑝 be a prime number and let ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and the completion of the algebraic closure of ℚ_𝑝, respectively. It is well known that the analogous of the classical gamma function Γ in 𝑝-adic context depends on modifying the factorial function 𝑛! [1]. The factorial function (𝑛!)_𝑝 in ℚ_𝑝 is defined as (𝑛!)_𝑝=_{𝑗<𝑛𝑝,𝑗=1}𝑗. (1) The 𝑝-adic gamma function Γ_𝑝 is defined by Morita [2] as the continuous extension to ℤ_𝑝 of the function 𝑛→(−1)^𝑛(𝑛!)_𝑝. That is, Γ_𝑝(𝑥) is defined by the formula Γ_𝑝(𝑥)=lim_𝑛→𝑥(−1)^𝑛_{𝑗<𝑛𝑝,𝑗=1}𝑗 (2) for 𝑥∈ℤ_𝑝, where 𝑛 approaches 𝑥 through positive integers. The 𝑝-adic gamma function Γ_𝑝(𝑥) had been studied by Diamond [3], Barsky [4], and others. The relationship between some special functions and the 𝑝-adic gamma function Γ_𝑝(𝑥) were investigated by Gross and Koblitz [5], Cohen and Friedman [6]. and Shapiro [7].

The 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞(𝑥) is defined by Koblitz as follows.

Definition 1 (see [8]). Let 𝑞∈ℂ_𝑝,|𝑞−1|_𝑝<1,𝑞≠1. The 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞(𝑥) is defined by formula Γ_𝑝,𝑞(𝑥)=lim_𝑛→𝑥(−1)^𝑛_{𝑗<𝑛(𝑝,𝑗)=1}1−𝑞^𝑗1−𝑞 (3) for 𝑥∈ℤ_𝑝, where 𝑛 approaches 𝑥 through positive integers. We recall that lim_𝑞→1Γ_𝑝,𝑞=Γ_𝑝.

The 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞(𝑥) was studied by Koblitz [8, 9], Nakazato [10], Kim et al. [11], and Kim [12].

2. Main Results

In the present work, we give a new identity for the 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞(𝑥) in special case 𝑝=2. Also, we derive some properties and representations for the 𝑞-extension of the 𝑝-adic gamma function Γ_𝑝,𝑞(𝑥).

Theorem 2. If 𝑝=2, then for all 𝑥∈ℤ₂Γ_2,𝑞(𝑥)Γ_2,𝑞(1−𝑥)=(−1)^{1+𝜎1(𝑥)}lim_{𝑛→𝑥𝑗<𝑛(2,𝑗)=1}𝑞^𝑗, (4) where 𝜎₁ is defined by the formula 𝜎₁^∞_𝑗=0𝑎_𝑗2^𝑗=𝑎₁. (5)

Proof. Let 𝑝=2 and 𝑛∈ℕ. From Proposition  3 in [12] we known that Γ_2,𝑞(𝑛+1)Γ_2,𝑞(−𝑛)=(−1)^{𝑛+1−[𝑛/2]}_{𝑗<𝑛+1(2,𝑗)=1}𝑞^𝑗. (6) Here, [⋅] is the greatest integer function. Taking 𝑛−1 in place of 𝑛, the relation becomes Γ_2,𝑞(𝑛)Γ_2,𝑞(1−𝑛)=(−1)^{𝑛−[(𝑛−1)/2]}_{𝑗<𝑛(2,𝑗)=1}𝑞^𝑗. (7) Now, let 𝑛=𝑎₀+𝑎₁2+𝑎₂2²+⋯ in base 2. If 𝑎₀≠0, then 𝑎₁=1 in base 2 and 𝑛−12=𝑎₀−1+𝑎₁2+𝑎₂2²+⋯2=𝑎₁+𝑎₂2+⋯≡𝑎₁(mod2). (8) Thus, we get (−1)^{𝑛−[(𝑛−1)/2]}=(−1)^𝑛(−1)^{−[(𝑛−1)/2]}=(−1)¹(−1)^−𝑎1=(−1)^1−𝑎1=(−1)^1+𝑎1=(−1)^1+𝜎1. (9) If 𝑎₀=0, then 𝑛−12=−1+𝑎₁2+𝑎₂2²+⋯2=1+𝑎₁−12+𝑎₂2²+⋯2≡𝑎₁−1(mod2). (10) Hence, (−1)^{𝑛−[(𝑛−1)/2]}=(−1)^𝑛(−1)^{−[(𝑛−1)/2]}=(−1)²(−1)^{−(𝑎1−1)}=(−1)^2+𝑎1−1=(−1)^1+𝑎1=(−1)^{1+𝜎1(𝑛)}. (11) Thus, we have Γ_2,𝑞(𝑛)Γ_2,𝑞(1−𝑛)=(−1)^{1+𝜎1(𝑛)}_{𝑗<𝑛(2,𝑗)=1}𝑞^𝑗 (12) and thus, we obtain Γ_2,𝑞(𝑥)Γ_2,𝑞(1−𝑥)=(−1)^{1+𝜎1(𝑥)}lim_{𝑛→𝑥𝑗<𝑛(2,𝑗)=1}𝑞^𝑗. (13)

We recall that the 𝑞-factorial [𝑛;𝑞]! is defined in [13] by the formula 𝑛;𝑞!=𝑛;𝑞𝑛−1;𝑞⋯2;𝑞1;𝑞 (14) for 𝑛≥1, where 𝑥;𝑞=1−𝑞^𝑥.1−𝑞 (15) Note that for 𝑛=0, we can define [0;𝑞]!=1.

We use the following theorem to prove our results.

Theorem 3 (see [12]). Let 𝑛∈ℕ. Then, Γ_𝑝,𝑞(𝑛+1)=(−1)^𝑛+1𝑛;𝑞!𝑝;𝑞^[𝑛/𝑝]𝑛/𝑝;𝑞^𝑝!, (16) where [⋅] is the greatest integer function. In particular, 𝑝^𝑛−1;𝑞!=(−1)^𝑝𝑝;𝑞^{𝑝𝑛−1−1}𝑝^𝑛−1−1;𝑞^𝑝!Γ_𝑝,𝑞𝑝^𝑛. (17)

Theorem 4. Let 𝑛∈ℕ and let 𝑠_𝑛 be the sum of the digits of 𝑛=∑^𝑠_𝑗=0𝑎_𝑗𝑝^𝑗(𝑎_𝑠≠0) in base 𝑝. Then (a)[[𝑛/𝑝^𝑠];𝑞]!=(−1)^{𝑛+1−𝑠}(−[𝑝;𝑞])^{(𝑛−𝑠𝑛)/(𝑝−1)}∏^𝑠−1_𝑗=0[[𝑛/𝑝^𝑗+1];𝑞^𝑝]!/[[𝑛/𝑝^𝑗];𝑞]!∏^𝑠_𝑗=0Γ_𝑝,𝑞([𝑛/𝑝^𝑗]+1)(b)[𝑛;𝑞]!=(−1)^{𝑛+1−𝑠}(−[𝑝;𝑞])^{(𝑛−𝑠𝑛)/(𝑝−1)}[[𝑛/𝑝];𝑞^𝑝]!∏^𝑠_𝑗=1[[𝑛/𝑝^𝑗+1];𝑞^𝑝]!/[[𝑛/𝑝^𝑗];𝑞]! ∏^𝑠_𝑗=0Γ_𝑝,𝑞([𝑛/𝑝^𝑗]+1).

Proof. From the Theorem 3 we know that 𝑛;𝑞!=(−1)^𝑛+1𝑝;𝑞^[𝑛/𝑝]𝑛𝑝;𝑞^𝑝!Γ_𝑝,𝑞(𝑛+1). (18) By taking [𝑛/𝑝⁰],[𝑛/𝑝¹],…,[𝑛/𝑝^𝑠] instead of 𝑛, respectively, we get the relations𝑛𝑝⁰;𝑞!=(−1)^{[𝑛/𝑝0]+1}𝑝;𝑞^[𝑛/𝑝1]×𝑛𝑝¹;𝑞^𝑝!Γ_𝑝,𝑞𝑛𝑝⁰,𝑛+1𝑝¹;𝑞!=(−1)^{[𝑛/𝑝1]+1}𝑝;𝑞^[𝑛/𝑝2]×𝑛𝑝²;𝑞^𝑝!Γ_𝑝,𝑞𝑛𝑝¹,⋮𝑛+1𝑝^𝑠;𝑞!=(−1)^{[𝑛/𝑝𝑠]+1}𝑝;𝑞^{[𝑛/𝑝𝑠+1]}×𝑛𝑝^𝑠+1;𝑞^𝑝!Γ_𝑝,𝑞𝑛𝑝^𝑠+1. (19) By multiplying of the equalities above, we can easily obtain 𝑛𝑝^𝑠;𝑞!=(−1)^{[𝑛/𝑝0]+⋯+[𝑛/𝑝𝑠]+𝑠+1}𝑝;𝑞^{[𝑛/𝑝1]+⋯+[𝑛/𝑝𝑠+1]}×𝑛𝑝^𝑠+1;𝑞^𝑝!^𝑠−1_𝑗=0𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗!×;𝑞_𝑠𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗=()+1−1^{(𝑛−𝑠𝑛)/(𝑝−1)}(−1)^{𝑛+1−𝑠}𝑝;𝑞^{(𝑛−𝑠𝑛)/(𝑝−1)}×𝑛𝑝^𝑠+1;𝑞^𝑝!^𝑠−1_𝑗=0𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗!×;𝑞_𝑠𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗+1. (20) Therefore, we get the relation (a) 𝑛𝑝^𝑠;𝑞!=(−1)^{𝑛+1−𝑠}−𝑝;𝑞^{(𝑛−𝑠𝑛)/(𝑝−1)}×^𝑠−1_𝑗=0𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗;𝑞!^𝑠_𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗,()+1𝑛;𝑞!=−1^{[𝑛/𝑝0]+⋯+[𝑛/𝑝𝑠]+𝑠+1}𝑝;𝑞^{[𝑛/𝑝1]+⋯+[𝑛/𝑝𝑠+1]}×𝑛𝑝;𝑞^𝑝!^𝑠_𝑗=1𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗!×;𝑞_𝑠𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗=()+1−1^{(𝑛−𝑠𝑛)/(𝑝−1)}(−1)^{𝑛+1−𝑠}𝑝;𝑞^{(𝑛−𝑠𝑛)/(𝑝−1)}×𝑛𝑝;𝑞^𝑝!^𝑠_𝑗=1𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗!×;𝑞_𝑠𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗+1. (21) Therefore, we get the relation (b) 𝑛;𝑞!=(−1)^{𝑛+1−𝑠}−𝑝;𝑞^{(𝑛−𝑠𝑛)/(𝑝−1)}𝑛𝑝;𝑞^𝑝!×^𝑠_𝑗=1𝑛/𝑝^𝑗+1;𝑞^𝑝!𝑛/𝑝^𝑗;𝑞!^𝑠_𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗+1. (22)

Theorem 5. Let 𝑛∈ℕ and let 𝑛=∑^𝑠_𝑗=0𝑎_𝑗𝑝^𝑗(𝑎_𝑠≠0). Then 𝑝^𝑛−1;𝑞!=(−1)^𝑝−𝑝;𝑞^{(𝑝𝑛−1)/(𝑝−1)}×𝑝;𝑞^−𝑛𝑝^𝑛−1−1;𝑞^𝑝!×^𝑛−2_𝑗=0𝑝^𝑗−1;𝑞^𝑝!𝑝^𝑗+1−1;𝑞!^𝑛_𝑗=0Γ_𝑝,𝑞𝑝^𝑗. (23)

Proof. From Theorem 3 it follows that 𝑝^𝑗−1;𝑞!=(−1)^𝑝𝑝;𝑞^{𝑝𝑗−1−1}𝑝^𝑗−1−1;𝑞^𝑝!Γ_𝑝,𝑞𝑝^𝑗. (24) Taking of 0,1,…,𝑛 instead of 𝑗, respectively, we have the equalities 𝑝⁰−1;𝑞!=1=(−1)Γ_𝑝,𝑞𝑝⁰,𝑝¹−1;𝑞!=(−1)^𝑝𝑝;𝑞^𝑝0−1𝑝⁰−1;𝑞^𝑝!Γ_𝑝,𝑞𝑝¹,𝑝²−1;𝑞!=(−1)^𝑝𝑝;𝑞^𝑝1−1𝑝¹−1;𝑞^𝑝!Γ_𝑝,𝑞𝑝²,⋮𝑝^𝑛−1;𝑞!=(−1)^𝑝𝑝;𝑞^{𝑝𝑛−1−1}𝑝^𝑛−1−1;𝑞^𝑝!Γ_𝑝,𝑞𝑝^𝑛. (25) By multiplying of the equalities above, we can easily obtain 𝑝^𝑛−1;𝑞!=(−1)^𝑛𝑝+1𝑝;𝑞^{𝑝0+𝑝1+⋯+𝑝𝑛−1−𝑛}𝑝^𝑛−1−1;𝑞^𝑝!×^𝑛−2_𝑗=0𝑝^𝑗−1;𝑞^𝑝!𝑝^𝑗+1−1;𝑞!^𝑛_𝑗=0Γ_𝑝,𝑞𝑝^𝑗. (26) Thus, 𝑝^𝑛−1;𝑞!=(−1)^𝑝−𝑝;𝑞^{(𝑝𝑛−1)/(𝑝−1)}×𝑝;𝑞^−𝑛𝑝^𝑛−1−1;𝑞^𝑝!×^𝑛−2_𝑗=0𝑝^𝑗−1;𝑞^𝑝!𝑝^𝑗+1−1;𝑞!^𝑛_𝑗=0Γ_𝑝,𝑞𝑝^𝑗. (27)

Lemma 6. Let 𝑛∈ℤ⁺, 𝑛=∑^𝑠_𝑗=0𝑎_𝑗𝑝^𝑗(𝑎_𝑠≠0), and let 𝑝 be a prime number. Then, for 𝑗=0,1,…,𝑠𝑛/𝑝^𝑗;𝑞!𝑝;𝑞^{[𝑛/𝑝𝑗]}𝑛/𝑝^𝑗;𝑞^𝑝!=^{[𝑛/𝑝𝑗]}_𝑘=11−𝑞^𝑘1−𝑞^𝑘𝑝(0≤𝑘≤𝑠). (28)

Proof. For 𝑗=0𝑛;𝑞!𝑝;𝑞^𝑛𝑛;𝑞^𝑝!=1;𝑞2;𝑞⋯𝑛,𝑞𝑝;𝑞^𝑛1;𝑞^𝑝2;𝑞^𝑝⋯𝑛;𝑞^𝑝=1−𝑞1−𝑞1−𝑞²⋯1−𝑞1−𝑞^𝑛×1−𝑞1−𝑞^𝑝1−𝑞^𝑛1−𝑞^𝑝1−𝑞^𝑝⋯1−𝑞^𝑛𝑝1−𝑞^𝑝−1=1−𝑞1−𝑞1−𝑞²⋯1−𝑞1−𝑞^𝑛×1−𝑞1−𝑞^𝑝1−𝑞1−𝑞^2𝑝⋯1−𝑞1−𝑞^𝑛𝑝1−𝑞⁻¹=1−𝑞1−𝑞²⋯1−𝑞^𝑛1−𝑞^𝑝1−𝑞^2𝑝⋯1−𝑞^𝑛𝑝. (29) For 1≤𝑗≤𝑠 it follows that 𝑛/𝑝^𝑗;𝑞!𝑝;𝑞^{[𝑛/𝑝𝑗]}𝑛/𝑝^𝑗;𝑞^𝑝!=1;𝑞2;𝑞⋯𝑛/𝑝^𝑗,𝑞𝑝;𝑞^{[𝑛/𝑝𝑗]}1;𝑞^𝑝2;𝑞^𝑝⋯𝑛/𝑝^𝑗;𝑞^𝑝=1−𝑞1−𝑞1−𝑞²⋯1−𝑞1−𝑞^{[𝑛/𝑝𝑗]}×1−𝑞1−𝑞^𝑝1−𝑞^{[𝑛/𝑝𝑗]}1−𝑞^𝑝1−𝑞^𝑝⋯1−𝑞^{[𝑛/𝑝𝑗]𝑝}1−𝑞^𝑝−1=1−𝑞1−𝑞²⋯1−𝑞^{[𝑛/𝑝𝑗]}1−𝑞^𝑝1−𝑞^2𝑝⋯1−𝑞^{[𝑛/𝑝𝑗]𝑝}. (30)
Then, we obtain 𝑛/𝑝^𝑗;𝑞!𝑝;𝑞^{[𝑛/𝑝𝑗]}𝑛/𝑝^𝑗;𝑞^𝑝!=^{[𝑛/𝑝𝑗]}_𝑘=11−𝑞^𝑘1−𝑞^𝑘𝑝. (31)

Theorem 7. Let 𝑛∈ℕ and let 𝑠_𝑛 be the sum of the digits of 𝑛=∑^𝑠_𝑗=0𝑎_𝑗𝑝^𝑗(𝑎_𝑠≠0) in base 𝑝. Then 𝑛;𝑞!=(−1)^{((𝑛−𝑠𝑛)/(𝑝−1))+𝑛+1−𝑠[𝑛/𝑝1]}_𝑘=11−𝑞^𝑘𝑝1−𝑞^𝑘_{[𝑛/𝑝𝑠]𝑘=1}⋯1−𝑞^𝑘𝑝1−𝑞^𝑘𝑠_𝑗=0Γ_𝑝,𝑞𝑛𝑝^𝑗+1. (32)

Proof. This theorem can be proved by using Theorem 4 and Lemma 6.

Acknowledgments

This work is supported by Mersin University and the Scientific and Technological Research Council of Turkey (TÜBİTAK). The authors would like to thank the reviewers for their useful comments and suggestions.