On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials
UDC 512.7Two bivariate kinds of $q$-Euler and $q$-Genocchi polynomials are introduced and their basic properties are stated and proved.  
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512244168654848 |
|---|---|
| author | Masjed-Jamei, M. Beyki , M. R. Masjed-Jamei, M. Beyki , M. R. |
| author_facet | Masjed-Jamei, M. Beyki , M. R. Masjed-Jamei, M. Beyki , M. R. |
| author_sort | Masjed-Jamei, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2025-03-31T08:49:21Z |
| description | UDC 512.7Two bivariate kinds of $q$-Euler and $q$-Genocchi polynomials are introduced and their basic properties are stated and proved.
  |
| doi_str_mv | 10.37863/umzh.v73i1.6039 |
| first_indexed | 2026-03-24T03:25:42Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i1.6039
UDC 512.7
M. Masjed-Jamei, M. R. Beyki (Dep. Math., K. N. Toosi Univ. Technology, Tehran, Iran)
ON A BIVARIATE KIND OF \bfitq -EULER AND \bfitq -GENOCCHI POLYNOMIALS*
ПРО БIВАРIАНТНI ПОЛIНОМИ ТИПУ \bfitq -ЕЙЛЕРА I \bfitq -ДЖЕНОКI
Two bivariate kinds of q-Euler and q-Genocchi polynomials are introduced and their basic properties are stated and proved.
Визначeно бiварiантнi полiноми типу q-Ейлера i q-Дженокi. Також cформульовано i доведено їхнi основнi власти-
востi.
1. Introduction. Euler and Genocchi polynomials have found valuable applications in various
branches of mathematics such as analytic number theory, numerical analysis, geometric design and
mathematical physics. For instance, Euler numbers are directly related to the Brouwer fixed point
theorem and vector fields [12]. These numbers are extended by Carlitz in [1] and called q-Euler
numbers. In [10], the authors have presented a new q-analogue of the exponential generating function
of Euler polynomials and in [5] a new q-extension of Euler numbers and polynomials are introduced.
In [2], the authors have obtained some new symmetric identities for q-Genocchi polynomials arising
from the fermionic p-adic q-integral on \BbbZ p. Finally, in [8], a new type of Euler polynomials and
numbers are introduced.
In this paper, we first give some preliminary definitions of q-calculus and the q-analogue of some
elementary functions, which are required in Section 3, in order to extend both ordinary q-Euler and
q-Genocchi polynomials. In this sense, we introduce a bivariate kind of q-Euler and q-Genocchi
polynomials in Section 3 and present some basic properties of the extended q-Euler polynomials.
Of course, because of similarity, we only give the properties of bivariate q-Genocchi polynomials
without proof in Section 4.
2. Preliminaries and definitions. If q \not = 1 and \alpha is a real number, the q-analogue of \alpha is
defined by [3, 4]
[\alpha ]q =
1 - q\alpha
1 - q
,
and
[n]q! =
n\prod
k=1
[k]q = [n]q[n - 1]q . . . [1]q, n \in \BbbN ,
is the q-analogue of n! where \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1[\alpha ]q = \alpha and [0]q! = 1.
The q-derivative operator of an arbitrary function defined by
Dqf(x) =
f(qx) - f(x)
(q - 1)x
,
satisfies the rules
* This work was supported by the Alexander von Humboldt Foundation (grant number Ref 3.4-IRN-1128637-GF-E).
c\bigcirc M. MASJED-JAMEI, M. R. BEYKI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1 77
78 M. MASJED-JAMEI, M. R. BEYKI
Dq
\bigl(
f(x)\pm kg(x)
\bigr)
= Dqf(x)\pm kDqg(x),
Dq
\bigl(
f(x)g(x)
\bigr)
= f(x)Dqg(x) + g(qx)Dqf(x) = g(x)Dqf(x) + f(qx)Dqg(x),
and
Dq
\biggl(
f(x)
g(x)
\biggr)
=
g(x)Dqf(x) - f(x)Dqg(x)
g(x)g(qx)
=
g(qx)Dqf(x) - f(qx)Dqg(x)
g(x)g(qx)
.
Although there is not a general chain rule for q-derivatives, we have
Dq
\bigl(
f(\alpha x\beta )
\bigr)
= \alpha [\beta ]qx
\beta - 1(Dq\beta f)(\alpha x
\beta )
and
Dq
\bigl(
f(\alpha x)
\bigr)
= \alpha (Dqf)(\alpha x).
The function
(x - a)nq =
\left\{ (x - a)(x - aq)(x - aq2) . . . (x - aqn - 1), n = 1, 2, . . . ,
1, n = 0,
(1)
is the q-analogue of (x - a)n, which can be extended to
(x - a) - n
q =
1
(x - aq - n)nq
, n \in \BbbN .
It is easy to check that Dq(x - a)nq = [n]q(x - a)n - 1
q .
The q-Pochhammer symbol is indeed a particular case of (1) for x = 1 and is defined as
(a; q)n =
n - 1\prod
k=0
(1 - aqk) with (a; q)0 = 1, n \in \BbbN . (2)
When n \rightarrow \infty , the limit relation of (2) is denoted by (a; q)\infty (provided that | q| < 1) and in the
sequel we have
(a; q)n =
(a; q)\infty
(aqn; q)\infty
, n \in \BbbN 0, | q| < 1,
while for any complex number \alpha , it reads as
(a; q)\alpha =
(a; q)\infty
(aq\alpha ; q)\infty
, | q| < 1.
The q-binomial coefficient is defined for positive integers n and k by\biggl[
n
k
\biggr]
q
=
[n]q!
[k]q![n - k]q!
=
(q; q)n
(q; q)k(q; q)n - k
=
\biggl[
n
n - k
\biggr]
q
.
In [9], Schork studied Ward’s “Calculus of sequences” and introduced a q-addition symbol as
(x\oplus q y)
n =
n\sum
k=0
\biggl[
n
k
\biggr]
q
xkyn - k.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ON A BIVARIATE KIND OF q-EULER AND q-GENOCCHI POLYNOMIALS 79
It is clear that the q-subtraction can be defined in the same way as
(x\ominus q y)
n =
n\sum
k=0
\biggl[
n
k
\biggr]
q
xk( - y)n - k =
\bigl(
x\oplus q ( - y)
\bigr) n
.
A q-analogue of the classical exponential function ex is defined by [3, 6]
Dqe
x
q = exq =
\infty \sum
n=0
xn
[n]q!
, 0 < | q| < 1, | x| < 1,
where
exqe
y
q = e
x\oplus qy
q and e
a(x\oplus qy)
q = e
ax\oplus qay
q .
Another q-type of the exponential function is defined by
Ex
q =
\infty \sum
n=0
q(
n
k)
xn
[n]q!
, 0 < | q| < 1,
so that these two q-exponential functions are closely related to each other by the relation
exqE
- x
q = 1. (3)
Finally, in this section we state q-Taylor’s theorem for formal power series [4].
Theorem 2.1. For any polynomial p(x) of degree n and any arbitrary point x = a, we have
p(x) =
n\sum
j=0
D(j)
q f(a)
(x - a)jq
[j]q!
.
Hence, any formal power series f(x) =
\sum \infty
j=0
cjx
j can be expressed in terms of a generalized
Taylor series
\sum \infty
j=0
D(j)
q f(0)
xj
[j]q!
such that
cj =
D
(j)
q f(0)
[j]q!
\forall j \in \BbbN 0 and Dqf(x) =
\infty \sum
j=1
[j]qcjx
j - 1.
2.1. \bfitq -Appell sets, \bfitq -Euler and \bfitq -Genocchi polynomials and some related properties. Let
\{ Pn(x)\} \infty n=0 be a polynomial set in which Pn(x) is of exact degree n. \{ Pn(x)\} \infty n=0 is a q-Appell
set if
DqPn+1(x) = [n+ 1]qPn(x).
Such sets were first introduced by Sharma and Chak [11]. The following characterization theorem
holds in this regard.
Theorem 2.2 [11]. Let \{ Pn(x)\} \infty n=0 ba a polynomial set. The following assertions are equiva-
lent:
1. \{ Pn(x)\} is a q-Appell polynomial set.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
80 M. MASJED-JAMEI, M. R. BEYKI
2. There exists a sequence (ak)k\geq 0 independent of n, a0 = 1, such that
Pn(x) =
n\sum
k=0
ak
[n]q!
[n - k]q!
xn - k.
3. \{ Pn(x)\} is generated by
A(t)eq(xt) =
\infty \sum
n=0
Pn(x)
tn
[n]q!
,
where
A(t) =
\infty \sum
k=0
ak
tk
[k]q!
, a0 = 1.
The q-Euler polynomials are defined by [5]
2extq
etq + 1
=
\infty \sum
n=0
En,q(x)
tn
[n]q!
,
leading to the representation
En,q(x) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Ek,q(0)x
n - k,
and the q-Genocchi polynomials are defined by [2]
2textq
etq + 1
=
\infty \sum
n=0
Gn,q(x)
tn
[n]q!
,
leading to the representation
Gn,q(x) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Gk,q(0)x
n - k.
It is not difficult to verify for every n \in \BbbN that
DqEn,q(x) = [n]qEn - 1,q(x) and DqGn,q(x) = [n]qGn - 1,q(x).
Hence, q-Euler and q-Genocchi polynomials belong to q-Appell set.
3. A bivariate kind of \bfitq -Euler polynomials. Let x, y \in \BbbR . Then the Taylor expansion of the
two functions ext \mathrm{c}\mathrm{o}\mathrm{s} yt and ext \mathrm{s}\mathrm{i}\mathrm{n} yt are respectively as follows [7]:
ext \mathrm{c}\mathrm{o}\mathrm{s} yt =
\infty \sum
k=0
Ck(x, y)
tk
k!
and
ext \mathrm{s}\mathrm{i}\mathrm{n} yt =
\infty \sum
k=0
Sk(x, y)
tk
k!
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ON A BIVARIATE KIND OF q-EULER AND q-GENOCCHI POLYNOMIALS 81
where
Ck(x, y) =
[ k2 ]\sum
j=0
( - 1)j
\biggl(
k
2j
\biggr)
xk - 2jy2j (4)
and
Sk(x, y) =
[ k - 1
2 ]\sum
j=0
( - 1)j
\biggl(
k
2j + 1
\biggr)
xk - 2j - 1y2j+1. (5)
Now, a q-extension of the bivariate polynomials (4) and (5) can be considered.
If x, y \in \BbbR , then
extq \mathrm{c}\mathrm{o}\mathrm{s}q yt =
\infty \sum
k=0
Ck,q(x, y)
tk
[k]q!
(6)
and
extq \mathrm{s}\mathrm{i}\mathrm{n}q yt =
\infty \sum
k=0
Sk,q(x, y)
tk
[k]q!
,
where
\mathrm{c}\mathrm{o}\mathrm{s}q z =
\infty \sum
n=0
( - 1)nz2n
[2n]q!
=
\infty \sum
n=0
1 + ( - 1)n
2
(\mathrm{i}z)n
[n]q!
and
\mathrm{s}\mathrm{i}\mathrm{n}q z =
\infty \sum
n=0
( - 1)nz2n+1
[2n+ 1]q!
= \mathrm{i}
\infty \sum
n=0
( - 1)n - 1
2
(\mathrm{i}z)n
[n]q!
.
In this sense, we have\Biggl( \infty \sum
k=0
ak
tk
[k]q!
\Biggr) \Biggl( \infty \sum
k=0
bk
tk
[k]q!
\Biggr)
=
\infty \sum
k=0
\left( k\sum
j=0
\biggl[
k
j
\biggr]
q
ajbk - j
\right) tk
[k]q!
. (7)
Proposition 3.1. The polynomials Ck,q(x, y) and Sk,q(x, y) can be explicitly represented as
Ck,q(x, y) =
[ k2 ]\sum
j=0
( - 1)j
\biggl[
k
2j
\biggr]
q
xk - 2jy2j (8)
and
Sk,q(x, y) =
[ k - 1
2 ]\sum
j=0
( - 1)j
\biggl[
k
2j + 1
\biggr]
q
xk - 2j - 1y2j+1. (9)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
82 M. MASJED-JAMEI, M. R. BEYKI
Proof. We have
extq \mathrm{c}\mathrm{o}\mathrm{s}q yt =
\Biggl( \infty \sum
k=0
(xt)k
[k]q!
\Biggr) \Biggl( \infty \sum
k=0
1 + ( - 1)k
2
(\mathrm{i}yt)k
[k]q!
\Biggr)
=
=
\infty \sum
k=0
\left( k\sum
j=0
\biggl[
k
j
\biggr]
q
1 + ( - 1)j
2
(\mathrm{i}y)jxk - j
\right) tk
[k]q!
=
=
\infty \sum
k=0
\left( [ k2 ]\sum
j=0
( - 1)j
\biggl[
k
2j
\biggr]
q
xk - 2jy2j
\right) tk
[k]q!
.
The proof of (9) is similar.
Proposition 3.2. The following derivative rules are valid:
Dq,xCk,q(x, y) = [k]qCk - 1,q(x, y), (10)
Dq,yCk,q(x, y) = - [k]qSk - 1,q(x, y), (11)
Dq,xSk,q(x, y) = [k]qSk - 1,q(x, y), (12)
and
Dq,ySk,q(x, y) = [k]qCk - 1,q(x, y). (13)
Proof. Relation (6) yields
\infty \sum
n=1
Dq,xCn,q(x, y)
tn
[n]q!
= textq \mathrm{c}\mathrm{o}\mathrm{s}q yt =
\infty \sum
n=0
Cn,q(x, y)
tn+1
[n]q!
=
=
\infty \sum
n=1
Cn - 1,q(x, y)
tn
[n - 1]q!
=
\infty \sum
n=0
[n]qCn - 1,q(x, y)
tn
[n]q!
,
proving (10). Other equations (11), (12) and (13) can be similarly derived.
Proposition 3.3. The following identities hold:
Ck,q(x, y) =
k\sum
j=0
\biggl[
k
j
\biggr]
q
Ck - j,q(0, y)x
j (14)
and
Sk,q(x, y) =
k\sum
j=0
\biggl[
k
j
\biggr]
q
Sk - j,q(0, y)x
j . (15)
Proof. By Proposition 3.2, for j = 0, 1, . . . , k we have
\partial j
q
\partial qxj
Ck,q(x, y) = [k]q[k - 1]q . . . [k - j + 1]qCk - j,q(x, y),
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ON A BIVARIATE KIND OF q-EULER AND q-GENOCCHI POLYNOMIALS 83
while for j > k we obtain
\partial j
q
\partial qxj
Ck,q(x, y) = 0,
because Ck,q(x, y) is a polynomial of degree k in terms of x. Hence, the q-Taylor expansion of
Ck,q(x, y) at x gives
Ck,q(x+ h, y) =
k\sum
j=0
\partial j
q
\partial qxj
Ck,q(x, y)
hj
[j]q!
=
k\sum
j=0
\biggl[
k
j
\biggr]
q
Ck - j(x, y)h
j ,
in which h \in \BbbR . It is now enough to take x = 0 and h = x to reach (14). In a similar way, (15) can
be derived.
Proposition 3.4. For any n \in \BbbN 0, the following power representations hold:
2n\sum
k=0
( - 1)n - kq(
k
2)
\biggl[
2n
k
\biggr]
q
C2n - k,q(x, y)x
k = y2n, (16)
2n+1\sum
k=0
( - 1)kq(
k
2)
\biggl[
2n+ 1
k
\biggr]
q
C2n+1 - k,q(x, y)x
k = 0, (17)
2n\sum
k=0
( - 1)kq(
k
2)
\biggl[
2n
k
\biggr]
q
S2n - k,q(x, y)x
k = 0, (18)
and
2n+1\sum
k=0
( - 1)n - kq(
k
2)
\biggl[
2n+ 1
k
\biggr]
q
S2n+1 - k,q(x, y)x
k = y2n+1. (19)
Proof. Multiplying both sides of (6) by E - xt
q and using (3), it follows that
\infty \sum
n=0
1 + ( - 1)n
2
\mathrm{i}nyn
tn
[n]q!
=
=
\Biggl( \infty \sum
n=0
q(
n
2)
[n]q!
( - 1)nxntn
\Biggr) \Biggl( \infty \sum
n=0
Cn,q(x, y)
tn
[n]q!
\Biggr)
=
=
\infty \sum
n=0
\Biggl(
n\sum
k=0
\biggl[
n
k
\biggr]
q
q(
k
2)( - 1)kxkCn - k,q(x, y)
\Biggr)
tn
[n]q!
.
By setting n \rightarrow 2n and n \rightarrow 2n+1 in the above relation, (16) and (17) are proved respectively. The
proof of (18) and (19) is similar.
Based on previous comments, we are now in a good position to introduce two kinds of bivariate
q-Euler polynomials as
2extq
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
\infty \sum
n=0
E(c)
n,q(x, y)
tn
[n]q!
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
84 M. MASJED-JAMEI, M. R. BEYKI
and
2extq
etq + 1
\mathrm{s}\mathrm{i}\mathrm{n}q yt =
\infty \sum
n=0
E(s)
n,q(x, y)
tn
[n]q!
,
and give some basic properties of them in the sequel.
Proposition 3.5. E
(c)
n,q(x, y) and E
(s)
n,q(x, y) can be represented in terms of q-Euler numbers as
follows:
E(c)
n,q(x, y) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Ek,q(0)Cn - k,q(x, y) (20)
and
E(s)
n,q(x, y) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Ek,q(0)Sn - k,q(x, y). (21)
Proof. By using the relation (7), we have
\infty \sum
n=0
E(c)
n,q(x, y)
tn
[n]q!
=
2
etq + 1
extq \mathrm{c}\mathrm{o}\mathrm{s}q yt =
=
\Biggl( \infty \sum
n=0
En,q(0)
tn
[n]q!
\Biggr) \Biggl( \infty \sum
n=0
Cn,q(x, y)
tn
[n]q!
\Biggr)
=
=
\infty \sum
n=0
\Biggl(
n\sum
k=0
\biggl[
n
k
\biggr]
q
Ek,q(0)Cn - k,q(x, y)
\Biggr)
tn
[n]q!
,
which proves (20). The proof of (21) is similar.
Proposition 3.6. E
(c)
n,q(x, y) and E
(s)
n,q(x, y) can be represented in terms of En,q(x) as follows:
E(c)
n,q(x, y) =
[n
2
]\sum
k=0
( - 1)k
\biggl[
n
2k
\biggr]
q
En - 2k,q(x)y
2k (22)
and
E(s)
n,q(x, y) =
[n - 1
2
]\sum
k=0
( - 1)k
\biggl[
n
2k + 1
\biggr]
q
En - 2k - 1,q(x)y
2k+1. (23)
Proof. The relation (22) follows since
\infty \sum
n=0
E(c)
n,q(x, y)
tn
[n]q!
=
2extq
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
=
\Biggl( \infty \sum
n=0
En,q(x)
tn
[n]q!
\Biggr) \Biggl( \infty \sum
n=0
1 + ( - 1)n
2
\mathrm{i}nyn
tn
[n]q!
\Biggr)
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ON A BIVARIATE KIND OF q-EULER AND q-GENOCCHI POLYNOMIALS 85
=
\infty \sum
n=0
\Biggl(
n\sum
k=0
\biggl[
n
k
\biggr]
q
1 + ( - 1)k
2
\mathrm{i}kykEn - k,q(x)
\Biggr)
tn
[n]q!
=
=
\infty \sum
n=0
\left( [n
2
]\sum
k=0
( - 1)k
\biggl[
n
2k
\biggr]
q
En - 2k,q(x)y
2k
\right) tn
[n]q!
.
Similarly, (23) can be proved.
Proposition 3.7. For every n \in \BbbN 0, the following identities hold:
E(c)
n,q
\bigl(
(1\oplus q x), y
\bigr)
+ E(c)
n,q(x, y) = 2Cn,q(x, y) (24)
and
E(s)
n,q
\bigl(
(1\oplus q x), y
\bigr)
+ E(s)
n,q(x, y) = 2Sn,q(x, y). (25)
Proof. We have
\infty \sum
n=0
E(c)
n,q
\bigl(
(1\oplus q x), y
\bigr) tn
[n]q!
=
2e
(1\oplus qx)t
q
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
=
2extq (etq + 1 - 1)
eq(t) + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt = 2extq \mathrm{c}\mathrm{o}\mathrm{s}q(yt) -
2extq
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
= 2
\infty \sum
n=0
Cn,q(x, y)
tn
[n]q!
-
\infty \sum
n=0
E(c)
n,q(x, y)
tn
[n]q!
,
which proves (24). The relation (25) can be similarly proved.
Corollary 3.1. The following relations hold:
E
(c)
2n,q(1, y) + E
(c)
2n,q(0, y) = 2( - 1)ny2n
and
E
(s)
2n+1,q(1, y) + E
(s)
2n+1,q(0, y) = 2( - 1)ny2n+1.
Proof. If n is replaced by 2n in (24) and x by 0, we obtain
E
(c)
2n,q(1, y) + E
(c)
2n,q(0, y) = 2C2n,q(0, y),
which proves the first relation because from (8) we have C2n,q(0, y) = ( - 1)ny2n. The second
relation can be similarly proved.
Proposition 3.8. For every n \in \BbbN , the following identities hold:
E(c)
n,q
\bigl(
(x\oplus q z), y
\bigr)
=
n\sum
k=0
\biggl[
n
k
\biggr]
q
E
(c)
k,q(x, y)z
n - k (26)
and
E(s)
n,q
\bigl(
(x\oplus q z), y
\bigr)
=
n\sum
k=0
\biggl[
n
k
\biggr]
q
E
(s)
k,q(x, y)z
n - k. (27)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
86 M. MASJED-JAMEI, M. R. BEYKI
Proof. We have
\infty \sum
n=0
E(c)
n,q
\bigl(
(x\oplus q z), y
\bigr) tn
[n]q!
=
2e
(x\oplus qz)t
q
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
=
2extq
etq + 1
eztq \mathrm{c}\mathrm{o}\mathrm{s}q yt =
\Biggl( \infty \sum
n=0
E(c)
n,q(x, y)
tn
[n]q!
\Biggr) \Biggl( \infty \sum
n=0
(zt)n
[n]q!
\Biggr)
=
=
\infty \sum
n=0
\Biggl(
n\sum
k=0
\biggl[
n
k
\biggr]
q
E
(c)
k,q(x, y)z
n - k
\Biggr)
tn
[n]q!
,
which proves (26). The proof of (27) is similar.
Corollary 3.2. For every n \in \BbbN , the following partial q-differential equations hold:
Dq,xE
(c)
n,q(x, y) = [n]qE
(c)
n - 1,q(x, y),
Dq,yE
(c)
n,q(x, y) = - [n]qE
(s)
n - 1,q(x, y),
Dq,xE
(s)
n,q(x, y) = [n]qE
(s)
n - 1,q(x, y),
and
Dq,yE
(s)
n,q(x, y) = [n]qE
(c)
n - 1,q(x, y).
4. A bivariate kind of \bfitq -Genocchi polynomials. In this section, we introduce a bivariate
kind of q-Genocchi polynomials and just present some basic propositions of them as their proofs are
similar to the previous section.
Based on pervious comments, we can introduce two kinds of bivariate q-Genocchi polynomials
as follows:
2textq
etq + 1
\mathrm{c}\mathrm{o}\mathrm{s}q yt =
\infty \sum
n=0
G(c)
n,q(x, y)
tn
[n]q!
and
2textq
etq + 1
\mathrm{s}\mathrm{i}\mathrm{n}q yt =
\infty \sum
n=0
G(s)
n,q(x, y)
tn
[n]q!
,
where they can be represented in terms of q-Genocchi numbers as
G(c)
n,q(x, y) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Gk,q(0)Cn - k,q(x, y)
and
G(s)
n,q(x, y) =
n\sum
k=0
\biggl[
n
k
\biggr]
q
Gk,q(0)Sn - k,q(x, y).
They can also be represented in terms of Gn,q(x) as follows:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ON A BIVARIATE KIND OF q-EULER AND q-GENOCCHI POLYNOMIALS 87
G(c)
n,q(x, y) =
[n
2
]\sum
k=0
( - 1)k
\biggl[
n
2k
\biggr]
q
Gn - 2k,q(x)y
2k
and
G(s)
n,q(x, y) =
[n - 1
2
]\sum
k=0
( - 1)k
\biggl[
n
2k + 1
\biggr]
q
Gn - 2k - 1,q(x)y
2k+1.
For every n \in \BbbN , the following identities hold:
G(c)
n,q
\bigl(
(1\oplus q x), y
\bigr)
+G(c)
n,q(x, y) = 2[n]qCn - 1,q(x, y)
and
G(s)
n,q
\bigl(
(1\oplus q x), y
\bigr)
+G(s)
n,q(x, y) = 2[n]qSn - 1,q(x, y).
Consequently, we have
G
(c)
2n+1,q(1, y) +G
(c)
2n,q(0, y) = 2[2n+ 1]q( - 1)ny2n
and
G
(s)
2n,q(1, y) +G
(s)
2n,q(0, y) = 2[2n]q( - 1)n+1y2n+1.
Moreover, for every n \in \BbbN ,
G(c)
n,q
\bigl(
(x\oplus q z), y
\bigr)
=
n\sum
k=0
\biggl[
n
k
\biggr]
q
G
(c)
k,q(x, y)z
n - k
and
G(s)
n,q
\bigl(
(x\oplus q z), y
\bigr)
=
n\sum
k=0
\biggl[
n
k
\biggr]
q
G
(s)
k,q(x, y)z
n - k.
Finally, for every n \in \BbbN , the following partial q-differential equations hold:
Dq,xG
(c)
n,q(x, y) = [n]qG
(c)
n - 1,q(x, y),
Dq,yG
(c)
n,q(x, y) = - [n]qG
(s)
n - 1,q(x, y),
Dq,xG
(s)
n,q(x, y) = [n]qG
(s)
n - 1,q(x, y),
and
Dq,yG
(s)
n,q(x, y) = [n]qG
(c)
n - 1,q(x, y).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
88 M. MASJED-JAMEI, M. R. BEYKI
References
1. L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76, 332 – 350 (1954).
2. U. Duran, M. Acikgoz, A. Esi, S. Araci, Some new symmetric identities involving q-Genocchi polynomials under S4 ,
J. Math. Anal., 6, 22 – 31 (2015).
3. T. Ernst, A comprehensive treatment of q-calculus, Birkhäuser (2012).
4. V. Kac, P. Cheung, Quantum calculus, Springer, 2001.
5. D. S. Kim, T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. and Appl.,
2014, № 1 (2014), 12 p.
6. T. Kim, q-Extension of the Euler formula and trigonometric functions, Russ. J. Math. Phys., 14, 275 – 278 (2007).
7. M. Masjed-Jamei, W. Koepf, Symbolic computation of some power-trigonometric series, J. Symbolic Comput., 80,
273 – 284 (2017).
8. M. Masjed-Jamei, M. R. Beyki, W. Koepf, A new type of Euler polynomials and numbers, Mediterr. J. Math., 15,
Article 138 (2018).
9. M. Schork, Wards ’calculus of sequences’ q-calculus and the limit q \rightarrow - 1, Adv. Stud. Contemp. Math., 13, 131 – 141
(2006).
10. J. Shareshian, M. L. Wachs, q-Eulerian polynomials: excedance number and major index, Electron. Res. Announc.
Amer. Math. Soc., 13, 33 – 45 (2007).
11. A. Sharma, A. Chak, The basic analogue of a class of polynomials, Riv. Math. Univ. Parma, 5, 325 – 337 (1954).
12. Y. Simsek, Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type
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2013-87.
Received 24.03.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
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| id | umjimathkievua-article-6039 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:25:42Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dc/9cc8263beb8a8211e138bafb93b70edc.pdf |
| spelling | umjimathkievua-article-60392025-03-31T08:49:21Z On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials Masjed-Jamei, M. Beyki , M. R. Masjed-Jamei, M. Beyki , M. R. Euler and Genocchi polynomials and numbers Appell polynomial set generating functions Euler and Genocchi polynomials and numbers Appell polynomial set generating functions UDC 512.7Two bivariate kinds of $q$-Euler and $q$-Genocchi polynomials are introduced and their basic properties are stated and proved. &nbsp; УДК 512.7 Про біваріантні поліноми типу $q$-Ейлера i $q$-Дженокі Визначeно біваріантні поліноми типу $q$-Ейлера i $q$-Дженокі. Також cформульовано і доведено їхні основні властивості. Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6039 10.37863/umzh.v73i1.6039 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 77 - 88 Український математичний журнал; Том 73 № 1 (2021); 77 - 88 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6039/8906 |
| spellingShingle | Masjed-Jamei, M. Beyki , M. R. Masjed-Jamei, M. Beyki , M. R. On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_alt | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_full | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_fullStr | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_full_unstemmed | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_short | On a bivariate kind of $q$-Euler and $q$-Genocchi polynomials |
| title_sort | on a bivariate kind of $q$-euler and $q$-genocchi polynomials |
| topic_facet | Euler and Genocchi polynomials and numbers Appell polynomial set generating functions Euler and Genocchi polynomials and numbers Appell polynomial set generating functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6039 |
| work_keys_str_mv | AT masjedjameim onabivariatekindofqeulerandqgenocchipolynomials AT beykimr onabivariatekindofqeulerandqgenocchipolynomials AT masjedjameim onabivariatekindofqeulerandqgenocchipolynomials AT beykimr onabivariatekindofqeulerandqgenocchipolynomials |