Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 176470, 4 pages
http://dx.doi.org/10.1155/2013/176470
Research Article
Some Properties of the 𝑞-Extension of the 𝑝-Adic
Gamma Function
Hamza Menken1 and Adviye Körükçü2
1
2
Mathematics Department, Science and Arts Faculty, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey
Mathematics Graduate Program, Institute of Science, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey
Correspondence should be addressed to Hamza Menken; hmenken@mersin.edu.tr
Received 10 October 2012; Accepted 7 February 2013
Academic Editor: Giovanni P. Galdi
Copyright © 2013 H. Menken and A. Körükçü. 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.
We study the 𝑞-extension of the 𝑝-adic gamma function Γ𝑝,𝑞 . We give a new identity for the 𝑞-extension of the 𝑝-adic gamma Γ𝑝,𝑞 in
the case 𝑝 = 2. Also, we derive some properties and new representations of the 𝑞-extension of the 𝑝-adic gamma Γ𝑝,𝑞 in general case.
1. Introduction
Let 𝑝 be a prime number and let Z𝑝 , Q𝑝 , and C𝑝 denote
the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and
the completion of the algebraic closure of Q𝑝 , 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 Q𝑝 is
defined as
(𝑛!)𝑝 = ∏ 𝑗.
𝑗<𝑛
(𝑝,𝑗)=1
(1)
The 𝑝-adic gamma function Γ𝑝 is defined by Morita [2] as the
continuous extension to Z𝑝 of the function 𝑛 → (−1)𝑛 (𝑛!)𝑝 .
That is, Γ𝑝 (𝑥) is defined by the formula
Γ𝑝 (𝑥) = lim (−1)𝑛 ∏ 𝑗
𝑛→𝑥
𝑗<𝑛
(𝑝,𝑗)=1
Definition 1 (see [8]). Let 𝑞 ∈ C𝑝 , |𝑞 − 1|𝑝 < 1, 𝑞 ≠ 1. The 𝑞extension of the 𝑝-adic gamma function Γ𝑝,𝑞 (𝑥) is defined by
formula
Γ𝑝,𝑞 (𝑥) = lim (−1)𝑛 ∏
𝑛→𝑥
𝑗<𝑛
(𝑝,𝑗)=1
1 − 𝑞𝑗
1−𝑞
(3)
for 𝑥 ∈ Z𝑝 , where 𝑛 approaches 𝑥 through positive integers.
We recall that lim𝑞 → 1 Γ𝑝,𝑞 = Γ𝑝 .
The 𝑞-extension of the 𝑝-adic gamma function Γ𝑝,𝑞 (𝑥)
was studied by Koblitz [8, 9], Nakazato [10], Kim et al. [11],
and Kim [12].
2. Main Results
(2)
for 𝑥 ∈ Z𝑝 , where 𝑛 approaches 𝑥 through positive
integers. The 𝑝-adic gamma function Γ𝑝 (𝑥) had been studied
by Diamond [3], Barsky [4], and others. The relationship
between some special functions and the 𝑝-adic gamma
function Γ𝑝 (𝑥) were investigated by Gross and Koblitz [5],
Cohen and Friedman [6]. and Shapiro [7].
The 𝑞-extension of the 𝑝-adic gamma function Γ𝑝,𝑞 (𝑥) is
defined by Koblitz as follows.
In the present work, we give a new identity for the 𝑞-extension
of the 𝑝-adic gamma function Γ𝑝,𝑞 (𝑥) in special case 𝑝 = 2.
Also, we derive some properties and representations for the
𝑞-extension of the 𝑝-adic gamma function Γ𝑝,𝑞 (𝑥).
Theorem 2. If 𝑝 = 2, then for all 𝑥 ∈ Z2
Γ2,𝑞 (𝑥) Γ2,𝑞 (1 − 𝑥) = (−1)1+𝜎1 (𝑥) lim ∏ 𝑞𝑗 ,
𝑛→𝑥
𝑗<𝑛
(2,𝑗)=1
(4)
2
Abstract and Applied Analysis
where 𝜎1 is defined by the formula
We recall that the 𝑞-factorial [𝑛; 𝑞]! is defined in [13] by
the formula
∞
𝜎1 ( ∑ 𝑎𝑗 2𝑗 ) = 𝑎1 .
(5)
[𝑛; 𝑞]! = [𝑛; 𝑞] [𝑛 − 1; 𝑞] ⋅ ⋅ ⋅ [2; 𝑞] [1; 𝑞]
𝑗=0
Proof. Let 𝑝 = 2 and 𝑛 ∈ N. From Proposition 3 in [12] we
known that
Γ2,𝑞 (𝑛 + 1) Γ2,𝑞 (−𝑛) = (−1)
𝑛+1−[𝑛/2]
for 𝑛 ≥ 1, where
[𝑥; 𝑞] =
𝑗
∏ 𝑞.
𝑗<𝑛+1
(2,𝑗)=1
(6)
Here, [⋅] is the greatest integer function. Taking 𝑛 − 1 in place
of 𝑛, the relation becomes
Γ2,𝑞 (𝑛) Γ2,𝑞 (1 − 𝑛) = (−1)𝑛−[(𝑛−1)/2] ∏ 𝑞𝑗 .
𝑗<𝑛
(2,𝑗)=1
[
𝑛−1
]= [
2
2
]
Γ𝑝,𝑞 (𝑛 + 1) = (−1)𝑛+1
= [𝑎1 + 𝑎2 2 + ⋅ ⋅ ⋅] ≡ 𝑎1 (mod 2) .
= (−1)
1−𝑎1
= (−1)
1+𝑎1
= (−1)
1+𝜎1
.
(9)
(−1 + 𝑎1 2 + 𝑎2 22 + ⋅ ⋅ ⋅)
𝑛−1
[
]= [
]
2
2
(1 + (𝑎1 − 1) 2 + 𝑎2 22 + ⋅ ⋅ ⋅)
2
,
(16)
where [⋅] is the greatest integer function. In particular,
[𝑝𝑛 − 1; 𝑞]! = (−1)𝑝 [𝑝; 𝑞]
[𝑝𝑛−1 − 1; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑝𝑛 ) .
(17)
(a) [[𝑛/𝑝𝑠 ]; 𝑞]! = (−1)𝑛+1−𝑠 (−[𝑝; 𝑞])(𝑛−𝑠𝑛 )/(𝑝−1)
𝑗+1
]; 𝑞𝑝 ]!/[[𝑛/𝑝𝑗 ]; 𝑞]!∏𝑠𝑗=0 Γ𝑝,𝑞 ([𝑛/𝑝𝑗 ] + 1)
∏𝑠−1
𝑗=0 [[𝑛/𝑝
Proof. From the Theorem 3 we know that
]
(10)
[𝑛/𝑝]
[𝑛; 𝑞]! = (−1)𝑛+1 [𝑝; 𝑞]
𝑛
[[ ] ; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑛 + 1) .
𝑝
(18)
By taking [𝑛/𝑝0 ], [𝑛/𝑝1 ], . . . , [𝑛/𝑝𝑠 ] instead of 𝑛, respectively,
we get the relations
≡ 𝑎1 − 1 (mod 2) .
Hence,
(−1)𝑛−[(𝑛−1)/2] = (−1)𝑛 (−1)−[(𝑛−1)/2]
[[
= (−1)2 (−1)−(𝑎1 −1)
= (−1)
[[𝑛/𝑝] ; 𝑞𝑝 ]!
(b) [𝑛; 𝑞]! = (−1)𝑛+1−𝑠 (−[𝑝; 𝑞])(𝑛−𝑠𝑛 )/(𝑝−1) [[𝑛/𝑝]; 𝑞𝑝 ]!
∏𝑠𝑗=1 [[𝑛/𝑝𝑗+1 ]; 𝑞𝑝 ]!/[[𝑛/𝑝𝑗 ]; 𝑞]!∏𝑠𝑗=0 Γ𝑝,𝑞 ([𝑛/𝑝𝑗 ] + 1).
If 𝑎0 = 0, then
= [
[𝑝; 𝑞]
Theorem 4. Let 𝑛 ∈ N and let 𝑠𝑛 be the sum of the digits of
𝑛 = ∑𝑠𝑗=0 𝑎𝑗 𝑝𝑗 (𝑎𝑠 ≠ 0) in base 𝑝. Then
Thus, we get
(−1)𝑛−[(𝑛−1)/2] = (−1)𝑛 (−1)−[(𝑛−1)/2] = (−1)1 (−1)−𝑎1
[𝑛; 𝑞]!
[𝑛/𝑝]
𝑝𝑛−1 −1
(8)
(15)
Theorem 3 (see [12]). Let 𝑛 ∈ N. Then,
2
(𝑎0 − 1 + 𝑎1 2 + 𝑎2 22 + ⋅ ⋅ ⋅)
1 − 𝑞𝑥
.
1−𝑞
Note that for 𝑛 = 0, we can define [0; 𝑞]! = 1.
We use the following theorem to prove our results.
(7)
Now, let 𝑛 = 𝑎0 +𝑎1 2+𝑎2 2 +⋅ ⋅ ⋅ in base 2. If 𝑎0 ≠ 0, then 𝑎1 = 1
in base 2 and
(14)
2+𝑎1 −1
× [[
(11)
= (−1)1+𝑎1
0
𝑛
[𝑛/𝑝1 ]
] ; 𝑞]! = (−1)[𝑛/𝑝 ]+1 [𝑝; 𝑞]
0
𝑝
[[
= (−1)1+𝜎1 (𝑛) .
1
𝑛
[𝑛/𝑝2 ]
] ; 𝑞]! = (−1)[𝑛/𝑝 ]+1 [𝑝; 𝑞]
1
𝑝
× [[
Thus, we have
Γ2,𝑞 (𝑛) Γ2,𝑞 (1 − 𝑛) = (−1)1+𝜎1 (𝑛) ∏ 𝑞𝑗
𝑗<𝑛
(2,𝑗)=1
[[
Γ2,𝑞 (𝑥) Γ2,𝑞 (1 − 𝑥) = (−1)1+𝜎1 (𝑥) lim ∏ 𝑞𝑗 .
𝑛→𝑥
𝑗<𝑛
(2,𝑗)=1
(13)
𝑛
𝑛
] ; 𝑞𝑝 ]!Γ𝑝,𝑞 ([ 1 ] + 1) ,
𝑝2
𝑝
..
.
(12)
and thus, we obtain
𝑛
𝑛
] ; 𝑞𝑝 ]!Γ𝑝,𝑞 ([ 0 ] + 1) ,
1
𝑝
𝑝
𝑠
𝑛
[𝑛/𝑝𝑠+1 ]
] ; 𝑞]! = (−1)[𝑛/𝑝 ]+1 [𝑝; 𝑞]
𝑠
𝑝
× [[
𝑛
𝑛
] ; 𝑞𝑝 ]!Γ𝑝,𝑞 ([ 𝑠 ] + 1) .
𝑝𝑠+1
𝑝
(19)
Abstract and Applied Analysis
3
By multiplying of the equalities above, we can easily obtain
[[
0
𝑠
𝑛
[𝑛/𝑝1 ]+⋅⋅⋅+[𝑛/𝑝𝑠+1 ]
] ; 𝑞]! = (−1)[𝑛/𝑝 ]+⋅⋅⋅+[𝑛/𝑝 ]+𝑠+1 [𝑝; 𝑞]
𝑠
𝑝
× [[
𝑗=0
𝑛−2
(𝑛−𝑠𝑛 )/(𝑝−1)
𝑗=0
𝑝𝑗−1 −1
𝑝
(20)
Therefore, we get the relation (a)
∏Γ𝑝,𝑞 ([
𝑗=0
𝑛
]+1) ,
𝑝𝑗
𝑝𝑛−1 −1
𝑗+1
By multiplying of the equalities above, we can easily obtain
𝑝0 +𝑝1 +⋅⋅⋅+𝑝𝑛−1 −𝑛
[𝑝𝑛 − 1; 𝑞]! = (−1)𝑛𝑝+1 [𝑝; 𝑞]
𝑛−2
𝑛
× ∏Γ𝑝,𝑞 ([ 𝑗 ] + 1)
𝑝
𝑗=0
×∏
− 1; 𝑞]! 𝑗=0
Thus,
[𝑝𝑛 − 1; 𝑞]! = (−1)𝑝 (− [𝑝; 𝑞])
× [𝑝; 𝑞]
𝑛−2
Therefore, we get the relation (b)
[𝑝𝑛−1 − 1; 𝑞𝑝 ]!
𝑗
𝑝
[𝑝 − 1; 𝑞 ]!
(27)
𝑛
Lemma 6. Let 𝑛 ∈ Z+ , 𝑛 = ∑𝑠𝑗=0 𝑎𝑗 𝑝𝑗 (𝑎𝑠 ≠ 0), and let 𝑝 be a
prime number. Then, for 𝑗 = 0, 1, . . . , 𝑠
𝑛
[[ ] ; 𝑞𝑝 ]!
𝑝
𝑛
×∏
∏Γ𝑝,𝑞 ([ 𝑗 ] + 1) .
𝑗 ] ; 𝑞]!
𝑝
[[𝑛/𝑝
𝑗=1
𝑗=0
−𝑛
(𝑝𝑛 −1)/(𝑝−1)
∏Γ𝑝,𝑞 (𝑝𝑗 ) .
𝑗+1 − 1; 𝑞]!
[𝑝
𝑗=0
𝑗=0
×∏
(21)
𝑠
𝑛
∏Γ𝑝,𝑞 (𝑝𝑗 ) .
(26)
𝑛
] + 1) .
𝑝𝑗
[[𝑛/𝑝𝑗+1 ] ; 𝑞𝑝 ]!
[𝑝𝑛−1 − 1; 𝑞𝑝 ]!
(𝑛−𝑠𝑛 )/(𝑝−1)
𝑠 [[𝑛/𝑝𝑗+1 ] ; 𝑞𝑝 ]!
𝑛
× [[ ] ; 𝑞𝑝 ]!∏
𝑗
𝑝
𝑗=1 [[𝑛/𝑝 ] ; 𝑞]!
𝑠
[𝑝𝑗 − 1; 𝑞𝑝 ]!
𝑗+1
𝑗=0 [𝑝
= (−1)(𝑛−𝑠𝑛 )/(𝑝−1) (−1)𝑛+1−𝑠 [𝑝; 𝑞]
(𝑛−𝑠𝑛 )/(𝑝−1)
[𝑝𝑛−1 − 1; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑝𝑛 ) .
(25)
𝑝
𝑠
[𝑛; 𝑞]! = (−1)𝑛+1−𝑠 (− [𝑝; 𝑞])
[𝑝1 − 1; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑝2 ) ,
[𝑝𝑛 − 1; 𝑞]! = (−1)𝑝 [𝑝; 𝑞]
[[𝑛/𝑝 ] ; 𝑞 ]!
𝑛
× [[ ] ; 𝑞𝑝 ]!∏
𝑗
𝑝
𝑗=1 [[𝑛/𝑝 ] ; 𝑞]!
𝑗=0
𝑝1 −1
[𝑛/𝑝1 ]+⋅⋅⋅+[𝑛/𝑝𝑠+1 ]
𝑠
𝑠
𝑠
[𝑝0 − 1; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑝1 ) ,
..
.
[𝑛; 𝑞]! = (−1)[𝑛/𝑝 ]+⋅⋅⋅+[𝑛/𝑝 ]+𝑠+1 [𝑝; 𝑞]
× ∏Γ𝑝,𝑞 ([
𝑝0 −1
[𝑝2 − 1; 𝑞]! = (−1)𝑝 [𝑝; 𝑞]
𝑛
(𝑛−𝑠 )/(𝑝−1)
[[ 𝑠 ] ; 𝑞]! = (−1)𝑛+1−𝑠 (−[𝑝; 𝑞] 𝑛
)
𝑝
𝑠−1 [[𝑛/𝑝𝑗+1 ] ; 𝑞𝑝 ]! 𝑠
[𝑝𝑗−1 − 1; 𝑞𝑝 ]!Γ𝑝,𝑞 (𝑝𝑗 ) .
(24)
[𝑝0 − 1; 𝑞]! = 1 = (−1) Γ𝑝,𝑞 (𝑝0 ) ,
[𝑝1 − 1; 𝑞]! = (−1)𝑝 [𝑝; 𝑞]
[[𝑛/𝑝𝑗 ] ; 𝑞]!
(23)
𝑛
Taking of 0, 1, . . . , 𝑛 instead of 𝑗, respectively, we have the
equalities
𝑛
] + 1) .
𝑝𝑗
0
[𝑝𝑗 − 1; 𝑞𝑝 ]!
[𝑝𝑗 − 1; 𝑞]! = (−1)𝑝 [𝑝; 𝑞]
𝑠−1 [[𝑛/𝑝
] ; 𝑞 ]!
𝑛
𝑝
]
;
𝑞
]!
∏
𝑠+1
𝑗
𝑝
𝑗=0 [[𝑛/𝑝 ] ; 𝑞]!
× ∏Γ𝑝,𝑞 ([
[𝑝𝑛−1 − 1; 𝑞𝑝 ]!
Proof. From Theorem 3 it follows that
𝑗+1
𝑠
(𝑝𝑛 −1)/(𝑝−1)
∏Γ𝑝,𝑞 (𝑝𝑗 ) .
𝑗+1 − 1; 𝑞]!
[𝑝
𝑗=0
𝑗=0
×∏
𝑛
] + 1)
𝑝𝑗
= (−1)(𝑛−𝑠𝑛 )/(𝑝−1) (−1)𝑛+1−𝑠 [𝑝; 𝑞]
𝑗=0
−𝑛
× [𝑝; 𝑞]
𝑛
] ; 𝑞𝑝 ]!∏
𝑗
𝑝𝑠+1
𝑗=0 [[𝑛/𝑝 ] ; 𝑞]!
𝑠
×∏
[𝑝𝑛 − 1; 𝑞]! = (−1)𝑝 (− [𝑝; 𝑞])
𝑠−1 [[𝑛/𝑝𝑗+1 ] ; 𝑞𝑝 ]!
× ∏Γ𝑝,𝑞 ([
× [[
Theorem 5. Let 𝑛 ∈ N and let 𝑛 = ∑𝑠𝑗=0 𝑎𝑗 𝑝𝑗 (𝑎𝑠 ≠ 0). Then
[[𝑛/𝑝𝑗 ] ; 𝑞]!
(22)
[𝑝; 𝑞]
[𝑛/𝑝𝑗 ]
[[𝑛/𝑝𝑗 ] ; 𝑞𝑝 ]!
[𝑛/𝑝𝑗 ]
= ∏
𝑘=1
1 − 𝑞𝑘
1 − 𝑞𝑘𝑝
(0 ≤ 𝑘 ≤ 𝑠) .
(28)
4
Abstract and Applied Analysis
Proof. For 𝑗 = 0
Acknowledgments
[𝑛; 𝑞]!
=
𝑛
[𝑝; 𝑞] [𝑛; 𝑞𝑝 ]!
[1; 𝑞] [2; 𝑞] ⋅ ⋅ ⋅ [𝑛, 𝑞]
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.
𝑛
[𝑝; 𝑞] [1; 𝑞𝑝 ] [2; 𝑞𝑝 ] ⋅ ⋅ ⋅ [𝑛; 𝑞𝑝 ]
=(
1 − 𝑞𝑛
1 − 𝑞 1 − 𝑞2
⋅⋅⋅
)
1−𝑞 1−𝑞
1−𝑞
𝑛
1 − 𝑞𝑝 1 − 𝑞𝑝
1 − 𝑞𝑛𝑝
× ((
⋅⋅⋅
)
)
𝑝
1−𝑞
1−𝑞
1 − 𝑞𝑝
=(
1 − 𝑞 1 − 𝑞2
1 − 𝑞𝑛
⋅⋅⋅
)
1−𝑞 1−𝑞
1−𝑞
×(
=
−1
1 − 𝑞𝑝 1 − 𝑞2𝑝
1 − 𝑞𝑛𝑝
⋅⋅⋅
)
1−𝑞 1−𝑞
1−𝑞
(1 − 𝑞) (1 − 𝑞2 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑛 )
(1 − 𝑞𝑝 ) (1 − 𝑞2𝑝 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑛𝑝 )
−1
.
(29)
For 1 ≤ 𝑗 ≤ 𝑠 it follows that
[[𝑛/𝑝𝑗 ] ; 𝑞]!
[𝑛/𝑝𝑗 ]
[𝑝; 𝑞]
=
[[𝑛/𝑝𝑗 ] ; 𝑞𝑝 ]!
[1; 𝑞] [2; 𝑞] ⋅ ⋅ ⋅ [[𝑛/𝑝𝑗 ] , 𝑞]
[𝑛/𝑝𝑗 ]
[𝑝; 𝑞]
[1; 𝑞𝑝 ] [2; 𝑞𝑝 ] ⋅ ⋅ ⋅ [[𝑛/𝑝𝑗 ] ; 𝑞𝑝 ]
𝑗
1 − 𝑞 1 − 𝑞2
1 − 𝑞[𝑛/𝑝 ]
=(
⋅⋅⋅
)
1−𝑞 1−𝑞
1−𝑞
[𝑛/𝑝𝑗 ]
1 − 𝑞𝑝
× ((
)
1−𝑞
(30)
𝑗
1 − 𝑞[𝑛/𝑝 ]𝑝
1 − 𝑞𝑝
⋅
⋅
⋅
)
1 − 𝑞𝑝
1 − 𝑞𝑝
−1
𝑗
=
(1 − 𝑞) (1 − 𝑞2 ) ⋅ ⋅ ⋅ (1 − 𝑞[𝑛/𝑝 ] )
𝑗
(1 − 𝑞𝑝 ) (1 − 𝑞2𝑝 ) ⋅ ⋅ ⋅ (1 − 𝑞[𝑛/𝑝 ]𝑝 )
.
Then, we obtain
[𝑛/𝑝𝑗 ]
[[𝑛/𝑝𝑗 ] ; 𝑞]!
[𝑛/𝑝𝑗 ]
[𝑝; 𝑞]
[[𝑛/𝑝𝑗 ] ; 𝑞𝑝 ]!
= ∏
𝑘=1
1 − 𝑞𝑘
.
1 − 𝑞𝑘𝑝
(31)
Theorem 7. Let 𝑛 ∈ N and let 𝑠𝑛 be the sum of the digits of
𝑛 = ∑𝑠𝑗=0 𝑎𝑗 𝑝𝑗 (𝑎𝑠 ≠ 0) in base 𝑝. Then
((𝑛−𝑠𝑛 )/(𝑝−1))+𝑛+1−𝑠
[𝑛; 𝑞]! = (−1)
[𝑛/𝑝1 ] (1
∏
𝑘=1
[𝑛/𝑝𝑠 ] (1
∏
𝑘=1
− 𝑞𝑘𝑝 )
(1 − 𝑞𝑘 )
𝑠
(1 − 𝑞𝑘 )
∏Γ𝑝,𝑞 ([
𝑗=0
− 𝑞𝑘𝑝 )
⋅⋅⋅
(32)
𝑛
] + 1) .
𝑝𝑗
Proof. This theorem can be proved by using Theorem 4 and
Lemma 6.
References
[1] W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic
Analysis, vol. 4, Cambridge University Press, Cambridge, UK,
1984.
[2] Y. Morita, “A 𝑝-adic analogue of the Γ-function,” Journal of the
Faculty of Science, vol. 22, no. 2, pp. 255–266, 1975.
[3] J. Diamond, “The 𝑝-adic log gamma function and 𝑝-adic Euler
constants,” Transactions of the American Mathematical Society,
vol. 233, pp. 321–337, 1977.
[4] D. Barsky, “On Morita’s 𝑝-adic gamma function,” Mathematical
Proceedings of the Cambridge Philosophical Society, vol. 89, no.
1, pp. 23–27, 1981.
[5] B. H. Gross and N. Koblitz, “Gauss sums and the 𝑝-adic Γfunction,” Annals of Mathematics, vol. 109, no. 3, pp. 569–581,
1979.
[6] H. Cohen and E. Friedman, “Raabe’s formula for 𝑝-adic gamma
and zeta functions,” Annales de l’Institut Fourier, vol. 58, no. 1,
pp. 363–376, 2008.
[7] I. Shapiro, “Frobenius map and the 𝑝-adic gamma function,”
Journal of Number Theory, vol. 132, no. 8, pp. 1770–1779, 2012.
[8] N. Koblitz, “q-extension of the p-adic gamma function,” Transactions of the American Mathematical Society, vol. 260, no. 2,
1980.
[9] N. Koblitz, “𝑞-extension of the 𝑝-adic gamma function. II,”
Transactions of the American Mathematical Society, vol. 273, no.
1, pp. 111–129, 1982.
[10] H. Nakazato, “The 𝑞-analogue of the 𝑝-adic gamma function,”
Kodai Mathematical Journal, vol. 11, no. 1, pp. 141–153, 1988.
[11] T. K. Kim et al., “A note on analogue of gamma functions,” in
Proceedings of the 5th Transcendental Number Theory, pp. 111–
118, Gakushin University, Tokyo, Japan, 1997.
[12] Y. S. Kim, “𝑞-analogues of 𝑝-adic gamma functions and 𝑝-adic
Euler constants,” Korean Mathematical Society, vol. 13, no. 4, pp.
735–741, 1998.
[13] R. Askey, “The 𝑞-gamma and 𝑞-beta functions,” Applicable
Analysis, vol. 8, no. 2, pp. 125–141, 1978/79.
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