R E S E A R C H
Open Access
On a
q
-analog of some numbers and
polynomials
Serkan Araci
1*, Erkan A ˘gyüz
2and Mehmet Acikgoz
2*Correspondence:
1Department of Economics, Faculty
of Economics, Administrative and Social Science, Hasan Kalyoncu University, Gaziantep, 27410, Turkey Full list of author information is available at the end of the article
Abstract
In this paper, we introduce newq-analogs of the Changhee numbers and polynomials of the first kind and of the second kind. We also derive some new interesting identities related to the Stirling numbers of the first kind and of the second kind, the Euler polynomials of higher order and theq-analogs of Euler polynomials by applying thep-adic integrals method and some summation transform techniques. It turns out that some well-known results are derived as special cases.
MSC: 05A19; 11B68; 11B83
Keywords: fermionicp-adicq-integral onZp; Changhee polynomials; Pochhammer symbol; Stirling numbers of the first kind; Stirling numbers of the second kind; higher order Euler polynomials
1 Introduction
In mathematics, special functions (or special polynomials) are known as ‘useful functions’. Because of their remarkable properties, special functions have been used for centuries. For instance, since they have numerous applications in astronomy, trigonometric functions have been studied for over a thousand years. Since then, the theory of special functions has been continuously developed with contributions by a host of mathematicians, including Euler, Legendre, Laplace, Gauss, Kummer, Eisenstein, Riemann, Ramanujan, and so on.
In the past years, the development of new special functions and of applications of special functions to new areas of mathematics have initiated a resurgence of interest in thep-adic analysis,q-analysis, analytic number theory, combinatorics, and so on. Moreover, in recent years, the various generalizations of the familiar special polynomials have been defined by usingp-adicq-integral onZpandp-adic fermionicq-deformed integrals onZpintroduced
and investigated by Kim [–]. Srivastava and Todorov [] derived an interesting extension of a representation for the generalized Bernoulli numbers in order to obtain interesting special cases considered earlier by Gould []. For more on these issues,e.g., see [–].
Letpbe chosen as a fixed odd prime number. Throughout this paper, we make use of the following notations.Zp denotes the ring of integers,Qp denotes the field ofp-adic
numbers, andCpdenotes the completion of the algebraic closure ofQp. Thep-adic norm | · |pis normalized by
|p|p=p–.
LetC(Zp) be the space of continuous functions onZp. Forf ∈C(Zp), the fermionicp
-adicq-invariant integral onZpis defined by Kim [, ], as follows:
I–q(f) =
Zp
f(x)dμ–q(x) =n→∞lim pn–
x=
f(x)(–q)
x
[pn]
–q
. (.)
It follows from (.) that
qI–q(f) +I–q(f) = []qf(), (.)
wheref(x) :=f(x+ ). Obviously
lim
q→I–q(f) =I–(f) =n→∞lim
pn–
x=
f(x)(–)x cf.[, ]. (.)
In [], the Changhee polynomials are defined by substitutingf(x) = ( +t)xinto (.)
with the case|t|p<p–
p–, as follows:
Zp
( +t)x+ydμ–(y) =
∞
n=
Zp
(x+y)ndμ–(y)
tn n!
=
∞
n=
Chn(x) tn n!
= +t( +t)
x, (.)
where (x)nis known as thePochhammer symbol(ordecreasing factorial) defined by
(x)n=x(x– )· · ·(x–n+ )
=
n
k=
S(n,k)xk (.)
and hereS(n,k) is the Stirling number of the first kind (see [–]).
In [], Kimet al.introduced usingp-adic integral techniques the idea that the Changee numbers are closely related to the Euler numbers as follows:
Em= m
n=
ChnS(n,m),
whereS(n,m) is the Stirling number of the second kind defined by the following
gener-ating series:
∞
n=m
S(n,m)
tn n! =
(et– )m
In [], Srivastava extended the Stirling numbers of the second kind toλ-Stirling num-bers of the second kind as follows:
(λet– )m m! =
∞
n=
S(n,m;λ)t
n
n!
m∈N∗andλ∈C
with, of course,
S(n,m; ) :=S(n,m).
In [], the Euler polynomials of (real or complex) orderα
E(nα)(x)
(sometimes called the Euler polynomials of higher order) are introduced by the following generating function:
et+
α
ext=
∞
n=
E(α)
n (x) tn n!
|t|<π (.)
with, of course,
E()n (x) :=En(x) and En(α)() :=E( α)
n ,
whereEn(x) andEn(α)are thenth Euler polynomials and thenth Euler numbers of orderα.
Recently, Kimet al.have studied the various generalizations of Changhee polynomials
cf.[, , ]. Ourq-analogs of the Changhee numbers and polynomials in the present paper are different from Kimet al.’sq-analogs of the Changhee numbers and polynomials. In this paper, we introduce aq-analog of the Changhee polynomials and derive some new interesting identities.
2 On aq-analog of Changhee numbers and polynomials
Let us now consider the followingp-adicq-integral representation in accordance with the Pochhammer symbol:
Zp
q–y(x+y)ndμ–q(y)
n∈Z+=N∪ {}
. (.)
From (.), we have
∞
n=
Zp
q–y(x+y)ndμ–q(y)
tn n! =
Zp
q–y
∞
n=
x+y n
tn dμ–q(y)
=
Zp
q–y( +t)x+ydμ–q(y), (.)
wheret∈Cpwith|t|p<p–
p–. Applying (.) to (.) gives
Zp
q–y( +t)x+ydμ–q(y) =
+q
+ ( +t)–( +t)
Let
Fq(x,t) =
+q
+ ( +t)–( +t)
x–.
Then
lim
q→Fq(x,t) =
+t( +t)
x.
Notice thatFq(x,t) seems to be a newq-extension of the generating function for
afore-mentioned Changhee polynomials of the first kind. Therefore, from (.) and (.), we obtain the following definition.
Definition LetFq(x,t) =
∞
n=Chn(x|q)t
n
n!, whereChn(x|q) is called aq-analog of the
nth Changhee polynomials of the first kind. Then we have forn≥
∞
n=
Chn(x|q) tn n! =
+q
+ ( +t)–( +t)
x–.
Moreover,
Chn(x|q) =
Zp
q–y(x+y)ndμ–q(y).
In the casex= in Definition , we haveChn(|q) :=Chn(q), which stands for theq
-analog of thenth Changhee numbers of the first kind. It follows from (.) and Definition that
+q
Chn(x) =Chn(x|q). (.)
Equation (.) shows that ourq-analog of the Changhee polynomials of the first kind is closely related to the Changhee polynomials. From (.) we have
Chn(q) = n
k=
S(n,k)Ek(q), (.)
whereEk(q) are theq-Euler polynomials derived from
Ek(q) =
Zp
q–yykdμ–q(y).
From (.), we have
Chn(x|q) =
Zp
q–y(x+y)ndμ–q(y)
=
n
k=
whereEk(x|q) are theq-Euler polynomials introduced by
Ek(x|q) =
Zp
q–y(x+y)kdμ–q(y).
From Definition , we have
+q qet+ =
∞
n=
Chn(q)
n!
et q –
n
=
∞
n=
Chn(q)
n!n!
∞
m=n
S(m,n)
(t–logq)m m!
=
∞
m=
m
n=
Chn(q)S(m,n)
(t–logq)m
m! . (.)
It follows from (.) that
∞
m=
+q
+qEm
qt m
m!=
∞
m=
m
n=
Chn(q)S(m,n)
tm m!.
When we compare the coefficientstnn! of both sides of the above we have
+q
+qEm
q=
m
n=
Chn(q)S(m,n).
Therefore, we obtain the following theorem.
Theorem For m≥,we have
+q
+qEm
q=
m
n=
Chn(q)S(m,n).
The increasing factorial sequence is known as
x(n)=x(x+ )(x+ )· · ·(x+n– ) n∈N∗.
For more information as regards this sequence, see [, , , , ].
Let us define theq-analog of the Changhee numbers of the second kind as follows:
Chn(q) =
Zp
q–y(–y)ndμ–q(y)
n∈N∗. (.)
It is easy to observe that
x(n)= (–)n(–x)n= n
k=
By virtue of (.) and (.), it leads to
Chn(q) =
Zp
q–y(–y)ndμ–q(y)
=
Zp
q–yy(n)(–)ndμ–q(y)
=
n
k=
S(n,k)(–)kEk(q). (.)
Thus, we state the following theorem.
Theorem The following holds true:
Chn(q) = n
k=
S(n,k)(–)kEk(q).
Let us now consider the generating function of theq-Changhee numbers of the second kind as follows:
∞
n=
Chn(q) tn n! =
∞
n=
Zp
q–y(–y)ndμ–q(y)
tn n!
=
Zp
q–y
∞
n=
–y
n
tn dμ–q(y)
=
Zp
q–y( +t)–ydμ–q(y), (.)
in whichZ pq
–y( +t)–ydμ
–q(y) equals
( +q)
+ ( +t)–. (.)
Then, combining (.) with (.), we state the following definition.
Definition Forn≥, we have
∞
n=
Chn(q) tn n! =
Zp
q–y( +t)–ydμ–q(y)
= ( +q) + ( +t)–.
Let us consider theq-Changhee polynomials of the second kind as follows:
+q
+t( +t)
–x= ∞
n=
Chn(x|q) tn
n!. (.)
Combining (.) with (.) at the valuex= , we have
Chn(|q) =
+q
It follows from (.) that
Zp
q–y( +t)–x–ydμ–q(y) = ∞
n=
Chn(x|q) tn
n!. (.)
From (.) gives
Chn(x|q) =
Zp
q–y(–x–y)ndμ–q(y)
=
n
k=
(–)kS(n,k)Ek(x|q) (n≥). (.)
Then, by (.), we have the following theorem.
Theorem The following holds true:
Chn(x|q) = n
k=
(–)kS(n,k)Ek(x|q) (n≥).
From (.) and (.), we have
q–x
+q qet+
e(–x)t=
∞
n=
Chn(x|q)
n!
qet– n
=
∞
n=
Chn(x|q) ∞
m=n
S(m,n;q)t
m
m!
=
∞
m=
m
n=
Chn(x|q)S(m,n;q) tm
m!. (.)
Further
q–x
Zp
e(–x+y)tdμ–q(y) = ∞
n=
Chn(x|q)
(qet– )n
n!
=
∞
m=
m
n=
Chn(x|q)S(m,n;q) tm
m!. (.)
Therefore, by (.) and (.), we obtain the following theorem.
Theorem The following equality holds true:
q–xEm( –x|q) = (–)nq–xEm
x|q–=
m
n=
Chn(x|q)S(m,n;q),
whereEm(x|q)may be called the mth q-Euler polynomials
Em(x|q) =
Zp
because
lim
q→Em(x|q) :=Em(x).
From Definition and (.) we have
(–)nChn(q)
n! = (–)
n
Zp
q–y
y n
dμ–q(y)
=
Zp
q–y
–y+n–
n
dμ–q(y)
=
n
m=
n–
n–m Zp
q–y
–y
m
dμ–q(y)
=
n
m=
n–
m–
Chm(q)
m! (.)
and
(–)nChn(q)
n! = (–)
n
Zp
q–y
–y
n
dμ–q(y)
=
Zp
q–y
y+n–
n
dμ–q(y)
=
n
m=
n–
n–m Zp
q–y
y m
dμ–q(x)
=
n
m=
n–
m–
Chm(q)
m! . (.)
Therefore, we get the following theorem.
Theorem The following holds:
(–)nChn(q)
n! =
n
m=
n–
m–
Chm(q) m!
and
(–)nChn(q)
n! =
n
m=
n–
m–
Chm(q) m! .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University,
Gaziantep, 27410, Turkey.2Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, Gaziantep, 27310, Turkey.
Acknowledgements
The authors are grateful to the comments of the referees, which have improved the quality of the paper substantially.
Received: 23 September 2014 Accepted: 22 December 2014 References
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