q Analogue of p adic log gamma functions associated with modified q extension of Genocchi numbers with weight alpha and beta

0.11 MB·English

by  Serkan Araci

q-ANALOGUE OF p-ADIC log Γ TYPE FUNCTIONS
ASSOCIATED WITH MODIFIED q-EXTENSION OF GENOCCHI
NUMBERS WITH WEIGHT α AND β
2
1
0 SERKANARACIANDMEHMETACIKGOZ
2

n
Abstract. The fundamental aim of this paper is to describe q-Analogue of
a
p-adic log gamma functions with weight alpha and beta. Moreover, we give
J
relationship between p-adic q-log gamma funtions with weight (α,β) and q-
0 extension of Genocchi numbers with weight alpha and beta and modified q-
2 Eulernumberswithweightα

]
T
N
1. Introduction
.
h
t Assume that p be a fixed odd prime number. Throughout this paper Z, Zp, Qp
a
and C will denote by the ring of integers,the field of p-adic rationalnumbers and
m p
the completion of the algebraic closure of Q , respectively. Also we denote N∗ =
p
[ N 0 andexp(x)=ex.Letvp :Cp Q (Q is the field of rational numbers)
∪{ } → ∪{∞}
2 denotethep-adicvaluationofCp normalizedsothatvp(p)=1. Theabsolutevalue
9v on Cp will be denoted as |.|p, and |x|p = p−vp(x) for x ∈ Cp. When one talks of
q-extensions, q is considered in many ways, e.g. as an indeterminate, a complex
0
number q C, or a p-adic number q C , If q C we assume that q < 1. If
3 p
∈ ∈ ∈ | |
1.1 tqh∈e Cfopll,owweinagssnuomtaeti|o1n−q|p <p−p−11, so that qx =exp(xlogq) for |x|p ≤1. We use
0 1 qx 1 ( q)x
2 (1.1) [x] = − , [x] = − −
q 1 q −q 1+q
1
−
v: where limq→1[x]q =x; cf. [1-24].
i For a fixed positive integer d with (d,f)=1, we set
X
X = X =limZ/dpNZ,
r d ←−
a N
X∗ = a+dpZ
p
0<a∪<dp
(a,p)=1
and
a+dpNZ = x X x a moddpN ,
p
∈ | ≡
where a Z satisfies the condition 0 a<dpN.
(cid:8) (cid:0) (cid:1)(cid:9)
∈ ≤
It is known that
qx
µ x+pNZ =
q p [pN]
q
(cid:0) (cid:1)
Date:January20,2012.
2000 Mathematics Subject Classification. Primary46A15,Secondary41A65.
Key words and phrases. Modifiedq-Genocchi numbers withweightalpha andbeta, Modified
q-Eulernumberswithweightalphaandbeta, p-adicloggammafunctions.
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2 SERKANARACIANDMEHMETACIKGOZ
is a distribution on X for q C with 1 q 1.
∈ p | − |p ≤
Let UD(Z ) be the set of uniformly differentiable function on Z . We say that
p p
f is a uniformly differentiable function at a point a Z , if the difference quotient
p
∈
f(x) f(y)
F (x,y)= −
f
x y
−
has a limit f´(a) as (x,y) (a,a) and denote this by f UD(Z ). The p-adic
p
→ ∈
q-integral of the function f UD(Z ) is defined by
p
∈
pN−1
1
(1.2) I (f)= f(x)dµ (x)= lim f(x)qx
q ZZp q N→∞[pN]q x=0
X
The bosonic integral is considered by Kim as the bosonic limit q 1, I (f) =
1
→
limq→1Iq(f). Similarly, the p-adic fermionic integration on Zp defined by Kim as
follows:
I−q(f)=q→lim−qIq(f)= Z f(x)dµ−q(x)
Z p
Let q 1, then we have p-adic fermionic integral on Z as follows:
p
→
pN−1
x
I−1(f)= lim Iq(f)= lim f(x)( 1) .
q→−1 N→∞ −
x=0
X
Stirling asymptotic series are defined by
Γ(x+1) 1 ∞ ( 1)n+1 B
n+1
(1.3) log = x logx+ − x
√2π − 2 n(n+1) xn −
(cid:18) (cid:19) (cid:18) (cid:19) n=1
X
where B are familiar n-th Bernoulli numbers cf. [6, 8, 9, 25].
n
Recently, Araci et al. defined modified q-Genocchi numbers and polynomials
with weight α and β in [4, 5] by the means of generating function:
∞
tn
(1.4) gn(α,q,β)(x) n! =t Z q−βξe[x+ξ]qαtdµ−qβ(ξ)
n=0 Z p
X
Sofromabove,weeasilygetWitt’sformulaofmodifiedq-Genocchinumbersand
polynomials with weight α and β as follows:
g(α,β) (x)
(1.5) nn+1+,q1 = Z q−βξ[x+ξ]nqαdµ−qβ (ξ)
Z p
where g(α,β)(0) := g(α,β) are modified q extension of Genocchi numbers with
n,q n,q
weight α and β cf. [4,5].
In [21], Rim and Jeong are defined modified q-Euler numbers with weight α as
follows:
(1.6) ξ(α) = q−t[t] dµ (t)
n,q qα −q
Z
Z p
e
From expressions of (1.5) and (1.6), we get the following Proposition 1:
ON THE q-ANALOGUE OF p-ADIC log GAMMA TYPE FUNCTIONS 3
Proposition 1. The following
(α) gn(α+,11),q
(1.7) ξ =
n,q n+1
is true.
e
In previous paper [6], Araci, Acikgoz and Park introduced weighted q-Analogue
of p-Adic log gamma type functions and they derived some interesting identities
in Analytic Numbers Theory and in p-Adic Analysis. They were motivated from
paperofT.Kimby”Ona q-analogueofthe p-adicloggammafunctionsandrelated
integrals, J. Number Theory, 76 (1999), no. 2, 320-329.” We also introduce q-
Analogue of p-Adic log gamma type function with weight α and β. We derive in
this paper some interesting identities this type of functions.
On p-adic log Γ function with weight α and β
In this part, from (1.2), we begin with the following nice identity:
n−1
(1.8) I(β) q−βxf +( 1)n−1I(β) q−βxf =[2] ( 1)n−1−lf(l)
−q n − −q qβ −
l=0
(cid:0) (cid:1) (cid:0) (cid:1) X
where f (x)=f(x+n) and n N (see [4]).
n
∈
In particular for n=1 into (1.8), we easily see that
(1.9) I(β) q−βxf +I(β) q−βxf =[2] f(0).
−q 1 −q qβ
With the simple appli(cid:0)cation, i(cid:1)t is easy(cid:0)to ind(cid:1)icate as follows:
∞ ( 1)n+1
(1.10) ((1+x)log(1+x))´=1+log(1+x)=1+ − xn
n(n+1)
n=1
X
where ((1+x)log(1+x))´= d ((1+x)log(1+x))
dx
By expression of (1.10), we can derive
∞ ( 1)n+1
(1.11) (1+x)log(1+x)= − xn+1+x+c, where c is constant.
n(n+1)
n=1
X
If we take x = 0, so we get c = 0. By expression of (1.10) and (1.11), we easily
see that,
∞ ( 1)n+1
(1.12) (1+x)log(1+x)= − xn+1+x.
n(n+1)
n=1
X
It is considered by T. Kim for q-analogue of p adic locally analytic function on
C Z as follows:
p p
\
(1.13) G (x)= [x+ξ] log[x+ξ] 1 dµ (ξ) (for detail, see[5,6]).
p,q Z q q− −q
Z p (cid:16) (cid:17)
By the same motivation of (1.13), in previous paper [6], q-analogue of p-adic
locally analytic function on C Z with weight α is considered
p p
\
(1.14) Gp(α,q)(x)= Z [x+ξ]qα log[x+ξ]qα −1 dµ−q(ξ)
Z p (cid:16) (cid:17)
(1)
In particular α=1 into (1.14), we easily see that, G (x)=G (x).
p,q p,q
4 SERKANARACIANDMEHMETACIKGOZ
Withthesamemanner,weintroduceq-Analogeofp-adiclocallyanalyticfunction
on C Z with weight α and β as follows:
p p
\
(1.15) G(α,β)(x)= q−βξ[x+ξ] log[x+ξ] 1 dµ (ξ)
p,q Z qα qα − −qβ
Z p (cid:16) (cid:17)
From expressions of (1.9) and (1.16), we state the following Theorem:
Theorem 1. The following identity holds:
G(α,β)(x+1)+G(α,β)(x)=[2] [x] log[x] 1 .
p,q p,q qβ qα qα −
(cid:16) (cid:17)
It is easy to show that,
1 qα(x+ξ)
(1.16) [x+ξ] = −
qα 1 qα
−
1 qαx+qαx qα(x+ξ)
= − −
1 qα
−
1 qαx 1 qαξ
= − +qαx −
1 qα 1 qα
(cid:18) − (cid:19) (cid:18) − (cid:19)
= [x] +qαx[ξ]
qα qα
Substituting x qαx[ξ]qα into (1.12) and by using (1.16), we get interesting
→ [x]qα
formula:
(1.17)
[x+ξ] log[x+ξ] 1 = [x] +qαx[ξ] log[x] + ∞ (−qαx)n+1[ξ]nqα+1 [x]
qα qα − qα qα qα n(n+1) [x]n − qα
(cid:16) (cid:17) (cid:16) (cid:17) nX=1 qα
Ifwesubstitute α=1into(1.17), wegetKim’sq-Analogueofp-adicloggamma
fuction (for detail, see[8]).
From expressionof (1.2) and (1.17), we obtain worthwhile and interesting theo-
rems as follows:
Theorem 2. For x C Z the following
p p
∈ \
(1.18)
G(α,β)(x)= [2]qβ [x] +qαxg2(α,q,β) log[x] + ∞ (−qαx)n+1 gn(α+,β1,)q [x] [2]qβ
p,q 2 qα 2 ! qα n=1n(n+1)(n+2) [x]nqα − qα 2
X
is true.
Corollary 1. Taking q 1 into (1.18), we get nice identity:
→
G ∞ ( 1)n+1 G
(α,β) 2 n+1
G (x)= x+ logx+ − x
p,1 2 n(n+1)(n+2) x −
(cid:18) (cid:19) n=1
X
where G are called famous Genocchi numbers.
n
Theorem 3. The following nice identity
(1.19)
G(α,1)(x)= [2]q [x] +qαxξ(α) log[x] + ∞ (−qαx)n+1 ξ(nα,q) [2]q [x]
p,q 2 qα 1,q qα n(n+1) [x]n − 2 qα
(cid:18) (cid:19) nX=1 e qα
is true. e
ON THE q-ANALOGUE OF p-ADIC log GAMMA TYPE FUNCTIONS 5
Corollary 2. Putting q 1 into (1.19), we have the following identity:
→
∞ ( 1)n+1 E
G(α,β)(x)=(x+E )logx+ − n x
p,1 1 n(n+1) xn −
n=1
X
where E are familiar Euler numbers.
n
References
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UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics,
27310Gaziantep,TURKEY
E-mail address: [e-mail]
UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics,
27310Gaziantep,TURKEY
E-mail address: [e-mail]