R E S E A R C H
Open Access
On a class of
q-Bernoulli and
q-Euler
polynomials
Nazim I Mahmudov
**Correspondence:
[email protected] Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, via Mersin 10, Turkey
Abstract
The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on theq-integers. The
q-analogues of well-known formulas are derived. Theq-analogue of the Srivastava-Pintér addition theorem is obtained. We give new identities involving
q-Bernstein polynomials.
1 Introduction
Throughout this paper, we always make use of the following notation:Ndenotes the set of natural numbers,Ndenotes the set of nonnegative integers,Rdenotes the set of real
numbers,Cdenotes the set of complex numbers. Theq-shifted factorial is defined by
(a;q)= , (a;q)n=
n–
j=
–qja, n∈N,
(a;q)∞= ∞
j=
–qja, |q|< ,a∈C.
Theq-numbers andq-numbers factorial are defined by
[a]q= –qa
–q (q= ); []q! = ; [n]q! = []q[]q· · ·[n]q, n∈N,a∈C,
respectively. Theq-polynomial coefficient is defined by
n k
q
= (q;q)n (q;q)n–k(q;q)k
.
Theq-analogue of the function (x+y)nis defined by
(x+y)nq:= n
k=
n k
q
qk(k–)xn–kyk, n∈N
.
Theq-binomial formula is known as
( –a)nq= (a;q)n= n–
j=
–qja= n
k=
n k
q
qk(k–)(–)kak.
In the standard approach to theq-calculus, two exponential functions are used
eq(z) = ∞
n=
zn [n]q!=
∞
k=
( – ( –q)qkz), <|q|< ,|z|<
| –q|,
Eq(z) = ∞
n=
qn(n–)zn
[n]q! =
∞
k=
+ ( –q)qkz, <|q|< ,z∈C.
From this form, we easily see thateq(z)Eq(–z) = . Moreover,
Dqeq(z) =eq(z), DqEq(z) =Eq(qz),
whereDqis defined by
Dqf(z) :=
f(qz) –f(z)
qz–z , <|q|< , =z∈C.
The aboveq-standard notation can be found in [].
Over years ago, Carlitz extended the classical Bernoulli and Euler numbers and poly-nomials and introduced theq-Bernoulli and theq-Euler numbers and polynomials (see [, ] and []). There are numerous recent investigations on this subject by, among many other authors, Cenkciet al.[–], Choiet al.[] and [], Kimet al.[–], Ozden and Simsek [], Ryooet al.[], Simsek [, ] and [], and Luo and Srivastava [], Srivas-tavaet al.[], Srivastava [], Mahmudov [].
We first give here the definitions of the q-Bernoulli and the q-Euler polynomials of higher order as follows.
Definition Letq,α∈C, <|q|< . Theq-Bernoulli numbersB(nα,q) and polynomials B(nα,q)(x,y) inx,yof orderαare defined by means of the generating functions:
t eq(t) –
α =
∞
n=
B(nα,q) t n
[n]q!, |t|< π,
t eq(t) –
α
eq(tx)Eq(ty) = ∞
n=
B(nα,q)(x,y) t n
[n]q!, |t|< π.
Definition Letq,α∈C, <|q|< . Theq-Euler numbersE(nα,q)and polynomialsE(nα,q() x,y)
inx,yof orderαare defined by means of the generating functions:
eq(t) + α
= ∞
n=
E(nα,q) t n
[n]q!, |t|<π,
eq(t) + α
eq(tx)Eq(ty) = ∞
n=
E(nα,q)(x,y) t n
It is obvious that
B(nα,q)=B(nα,q)(, ), lim q→–B
(α)
n,q(x,y) =B( α)
n (x+y), q→lim–B
(α)
n,q=B( α)
n ,
E(nα,q)=E(nα,q)(, ), lim q→–E
(α)
n,q(x,y) =En(α)(x+y), q→lim
–E
(α)
n,q=E(nα), lim
q→–B
(α)
n,q(x, ) =B( α)
n (x), q→lim–B
(α)
n,q(,y) =B( α)
n (y),
lim q→–E
(α)
n,q(x, ) =E( α)
n (x), q→lim–E
(α)
n,q(,y) =E( α)
n (y).
HereB(nα)(x) andE(nα)(x) denote the classical Bernoulli and Euler polynomials of orderα which are defined by
t et–
α
etx= ∞
n=
B(nα)(x)t n
n! and
et+ α
etx= ∞
n=
E(nα)(x)t n
n!.
In fact, Definitions and define two different types B(nα,q() x, ) and B(nα,q(,) y) of the
q-Bernoulli polynomials and two different types E(nα,q() x, ) and E(nα,q(,) y) of the q-Euler
polynomials. Both polynomialsB(nα,q() x, ) andB(nα,q(,) y) (E(nα,q() x, ) andE(nα,q(,) y)) coincide
with the classical higher-order Bernoulli polynomials (Euler polynomials) in the limiting caseq→–.
For theq-Bernoulli numbersBn,qand theq-Euler numbersEn,qof ordern, we have
Bn,q=Bn,q(, ) =B()n,q(, ), En,q=En,q(, ) =E()n,q(, ),
respectively. Note that theq-Bernoulli numbersBn,qare defined and studied in []. The aim of the present paper is to obtain some results for the above newly defined
q-Bernoulli andq-Euler polynomials. It should be mentioned thatq-Bernoulli andq-Euler polynomials in our definitions are polynomials ofxandy, and wheny= , they are polyno-mials ofx, but in other definitions they are functions ofqx. First advantage of this approach is that forq→–,B(α)
n,q(x,y) (E(nα,q() x,y)) becomes the classical BernoulliB(nα)(x+y) (Euler E(nα)(x+y)) polynomial, and we may obtain theq-analogues of well-known results, for ex-ample, those of Srivastava and Pintér [], Cheon [],etc.The second advantage is that we find the relation betweenq-Bernstein polynomials and Phillipsq-Bernoulli polynomials and derive the formulas involving theq-Stirling numbers of the second kind,q-Bernoulli polynomials and Phillipsq-Bernstein polynomials.
2 Preliminaries and lemmas
In this section we provide some basic formulas for theq-Bernoulli andq-Euler polynomi-als in order to obtain the main results of this paper in the next section. The following result is aq-analogue of the addition theorem for the classical Bernoulli and Euler polynomials.
Lemma (Addition theorems) For all x,y∈C,we have
B(nα,q)(x,y) = n
k=
n k
q
Bk(α,q)(x+y)qn–k, E(nα,q)(x,y) = n
k=
n k
q
B(nα,q)(x,y) = n
k=
n k
q
q(n–k)(n–k–)/Bk(α,q)(x, )yn–k= n
k=
n k
q
B(kα,q)(,y)xn–k, ()
E(nα,q)(x,y) = n
k=
n k
q
q(n–k)(n–k–)/Ek(α,q)(x, )yn–k= n
k=
n k
q
E(kα,q)(,y)xn–k. ()
In particular, settingx= andy= in () and (), we get the following formulas for
q-Bernoulli andq-Euler polynomials, respectively
B(nα,q)(x, ) = n
k=
n k
q
B(kα,q)xn–k, B(nα,q)(,y) = n
k=
n k
q
q(n–k)(n–k–)/Bk(α,q)yn–k, ()
E(nα,q)(x, ) = n
k=
n k
q
E(kα,q)xn–k, E(nα,q)(,y) = n
k=
n k
q
q(n–k)(n–k–)/Ek(α,q)yn–k. ()
Settingy= andx= in () and (), we get
B(nα,q)(x, ) = n
k=
n k
q
q(n–k)(n–k–)/Bk(α,q)(x, ), B(nα,q)(,y) = n
k=
n k
q
B(kα,q)(,y), ()
E(nα,q)(x, ) = n
k=
n k
q
q(n–k)(n–k–)/Ek(α,q)(x, ), E(nα,q)(,y) = n
k=
n k
q
E(kα,q)(,y). ()
Clearly, () and () areq-analogues of
B(nα)(x+ ) = n
k=
n k
B(kα)(x), E(nα)(x+ ) = n
k=
n k
E(kα)(x),
respectively.
Lemma We have
Dq,xB(nα,q)(x,y) = [n]qB(nα–,) q(x,y), Dq,yB(nα,q)(x,y) = [n]qB(nα–,) q(x,qy),
Dq,xE(nα,q)(x,y) = [n]qE( α)
n–,q(x,y), Dq,yEn(α,q)(x,y) = [n]qE( α)
n–,q(x,qy).
Lemma (Difference equations) We have
Bn(α,q)(,y) –B(nα,q)(,y) = [n]qBn(α–,–)q(,y), () En(α,q)(,y) +En(α,q)(,y) = E(nα,q–)(,y), () Bn(α,q)(x, ) –B(nα,q)(x, –) = [n]qB(nα–,–)q(x, –),
En(α,q)(x, ) +En(α,q)(x, –) = E(nα,q–)(x, –).
From () and (), () and (), we obtain the following formulas.
Lemma We have
B(nα,q–)(,y) = [n+ ]q
n
k=
n+
k
q
E(nα,q–)(,y) = we arrive at the following expansions:
yn=
which areq-analogues of the following familiar expansions:
yn=
following difference relationships:
3 Explicit relationship between theq-Bernoulli andq-Euler polynomials
In this section we investigate some explicit relationships between the q-Bernoulli and
q-Euler polynomials. Here someq-analogues of known results are given. We also obtain new formulas and some of their special cases below. These formulas are some extensions of the formulas of Srivastava and Pintér, Cheon and others.
We present natural q-extensions of the main results in the papers [] and [], see Theorems and .
Theorem For n∈N,the following relationships hold true:
B(nα,q)(x,y) =
Proof Using the following identity:
It is clear that
On the other hand,
It remains to use the formula ().
Next we discuss some special cases of Theorem .
Corollary For n∈N,m∈N,the relationship
holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary For n∈N,the following relationship holds true:
Corollary For n∈N,the following relationship holds true:
Bn,q(x, ) =
The formulas ()-() are theq-extension of Cheon’s main result []. Notice thatB,q= –[]q , see [], and the extra term becomes zero forq→–.
hold true between the q-Bernoulli polynomials and q-Euler polynomials.
Proof The proof is based on the following identities:
and is similar to that of Theorem .
Corollary For n∈N,m∈N,the relationship
En,q(x,y) =
n
k=
n k
q
m–n [k+ ]q
k+
j=
k+
j
q
mj(x– )jq
– k+
j=
k+
j
q
mjEj,q(x, –) –mk+Ek+,q(x, )
Bn–k,q(,my)
holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary ([]) For n∈N,m∈N,the following relationships hold true:
En(x+y) = n
k=
k+
n k
yk+–Ek+(y)
Bn–k(x),
En(x+y) = n
k=
n k
mk–n+ k+
x+ –m
m
k+
–Ek+
x+ –m
m
–Ek+(x)
Bn–k(my).
Corollary For n∈N,the following relationship holds true:
En,q(x,y) =
n
k=
n k
q [k+ ]q
qk(k+)yk+–E
k+,q(,y)
Bn–k,q(x, ).
Corollary For n∈N,the following relationships hold true:
En,q(x, ) = – n
k=
n k
[k+ ]qEk+,qBn–k,q(x, ),
En,q(,y) = –
n
k=
n k
q [k+ ]q
Ek+,qBn–k,q(,y).
These formulas areq-analogues of the formula of Srivastava and Pintér [].
4 q-Stirling numbers andq-Bernoulli polynomials
In this section, we aim to derive several formulas involving theq-Bernoulli polynomials, the q-Euler polynomials of orderα, theq-Stirling numbers of the second kind and the
q-Bernstein polynomials.
Theorem Each of the following relationships holds true for the Stirling numbers S(n,k)
of the second kind:
B(nα,q)(x,y) = n
j=
mx
j
j! n–j
k=
n k
q
mj–nB(kα,q)(,y)S(n–k,j),
E(nα,q)(x,y) = n
j=
mx
j
j! n–j
k=
n k
q
The familiarq-Stirling numbersS(n,k) of the second kind are defined by
(eq(t) – )k [k]q! =
∞
m=
S,q(m,k)
tm [m]q!,
wherek∈N. Next we give the relationship betweenq-Bernstein basis defined by Phillips [] andq-Bernoulli polynomials
bn,k(q;x) :=
n k
q
xk( –x)nq–k.
Theorem We have
bn,k(q;x) =xk n
m=
n m
q
S,q(m,k)Bn(k–)m,q(, –x). ()
Proof The proof follows from the following identities:
xktk
[k]q!eq(t)Eq(–xt) =
xktk [k]q!
∞
n=
( –x)n qtn [n]q! =
∞
n=k
n k
q
xk( –x)n–k q tn [n]q!
= ∞
n=k
bn,k(q;x)
tn [n]q!
and
xktk
[k]q!eq(t)Eq(–xt) =
xk(e
q(t) – )k [k]q!
tk
(eq(t) – )keq(t)Eq(–xt)
=xk
∞
m=
S,q(m,k)
tm [m]q!
∞
n=
B(nk,)q(, –x) t n
[n]q!
=xk
∞
n=
n
m=
n m
q
S,q(m,k)Bn(k–)m,q(, –x)
tn [n]q!
.
Finally, in their limit case whenq→–, this last result () would reduce to the following formula for the classical Bernoulli polynomialsB(nk)(x) and the Bernstein basisbn,k(x) =
n k
xk( –x)n–k:
bn,k(x) =xk
n
m=
n m
S(m,k)Bn(k–)m( –x).
Competing interests
The author declares that he has no competing interests.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have helped improve this article.
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Cite this article as:Mahmudov:On a class ofq-Bernoulli andq-Euler polynomials.Advances in Difference Equations
