A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials
UDC 517.5 We introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function.  These new sequences are generalizations of the poly-Euler numbers and polynomials.  We give several combinatorial identities and properties of t...
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| author | Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. |
| author_facet | Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. |
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| datestamp_date | 2022-03-26T11:01:37Z |
| description | UDC 517.5
We introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function.  These new sequences are generalizations of the poly-Euler numbers and polynomials.  We give several combinatorial identities and properties of these new polynomials, and also show some relations with $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue. |
| doi_str_mv | 10.37863/umzh.v72i4.6048 |
| first_indexed | 2026-03-24T03:25:51Z |
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DOI: 10.37863/umzh.v72i4.6048
UDC 517.5
T. Komatsu (Dep. Math., School Sci., Zhejiang Sci.-Tech. Univ., Hangzhou, China),
J. L. Ramı́rez* (Univ. Nac. Colombia, Bogotá),
V. F. Sirvent (Univ. Católica del Norte, Antofagasta, Chile)
A (\bfitp , \bfitq )-ANALOGUE OF POLY-EULER POLYNOMIALS
AND SOME RELATED POLYNOMIALS
(\bfitp , \bfitq )-АНАЛОГ ПОЛIЕЙЛЕРIВСЬКИХ ПОЛIНОМIВ
ТА ДЕЯКI СУМIЖНI ПОЛIНОМИ
We introduce a (p, q)-analogue of the poly-Euler polynomials and numbers by using the (p, q)-polylogarithm function.
These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial
identities and properties of these new polynomials, and also show some relations with (p, q)-poly-Bernoulli polynomials
and (p, q)-poly-Cauchy polynomials. The (p, q)-analogues generalize the well-known concept of the q-analogue.
Введено (p, q)-аналоги полiейлерiвських полiномiв i чисел за допомогою (p, q)-полiлогарифмiчної функцiї, якi є
узагальненнями полiейлерiвських полiномiв i чисел. Знайдено властивостi цих полiномiв i наведено деякi вiдпо-
вiднi комбiнаторнi рiвностi. Також показано зв’язок iз (p, q)-полiномами типу Бернуллi та Кошi. Цi (p, q)-аналоги
узагальнюють вiдому концепцiю q-аналогiв.
1. Introduction. The Euler numbers are defined by the generating function
2
et + e - t
=
\infty \sum
n=0
En
tn
n!
.
The sequence (En)n counts the numbers of alternating n-permutations. A n-permutation \sigma is
alternating if the n - 1 differences \sigma (i+1) - \sigma (i) for i = 1, 2, . . . , n - 1 have alternating signs. For
example, (1324) and (3241) are alternating permutations (cf. [10]).
The Euler polynomials are given by the generating function
2ext
et + 1
=
\infty \sum
n=0
En(x)
tn
n!
.
Note that En = 2nEn(1/2).
Many kinds of generalizations of these numbers and polynomials have been presented in the
literature (see, e.g., [39]). In particular, we are interested in the poly-Euler numbers and polynomials
(cf. [12, 15, 16, 32]).
The poly-Euler polynomials E
(k)
n (x) are defined by the following generating function:
2Lik(1 - e - t)
1 + et
ext =
\infty \sum
n=0
E(k)
n (x)
tn
n!
, k \in \BbbZ ,
where
* The research of J. Ramı́rez was partially supported by Universidad Nacional de Colombia (project No. 37805).
c\bigcirc T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4 467
468 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
Lik(t) =
\infty \sum
n=1
tn
nk
(1)
is the kth polylogarithm function. Note that if k = 1, then Li1(t) = - \mathrm{l}\mathrm{o}\mathrm{g}(1 - t), therefore,
E
(1)
n (x) = En - 1(x) for n \geq 1.
It is also possible to define the poly-Bernoulli and poly-Cauchy numbers and polynomials from
the kth polylogarithm function. In particular, the poly-Bernoulli numbers B
(k)
n were introduced by
Kaneko [17] by using the following generating function:
Lik(1 - e - t)
1 - e - t
=
\infty \sum
n=0
B(k)
n
tn
n!
, k \in \BbbZ .
If k = 1 we get B(1)
n = ( - 1)nBn for n \geq 0, where Bn are the Bernoulli numbers. Remember that
the Bernoulli numbers Bn are defined by the generating function
t
et - 1
=
\infty \sum
n=0
Bn
tn
n!
.
The poly-Bernoulli numbers and polynomials have been studied in several papers; among other
references, see [3, 4, 7, 8, 21, 22, 25 – 27].
The poly-Cauchy numbers of the first kind c
(k)
n were introduced by the first author in [19]. They
are defined as follows:
c(k)n =
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
(t1 . . . tk)n dt1 . . . dtk,
where (x)n = x(x - 1) . . . (x - n+ 1)(n \geq 1) with (x)0 = 1. Moreover, its exponential generating
function is
Lifk(\mathrm{l}\mathrm{n}(1 + t)) =
\infty \sum
n=0
c(k)n
tn
n!
, k \in \BbbZ ,
where
Lifk(t) =
\infty \sum
n=0
tn
n!(n+ 1)k
is the kth polylogarithm factorial function. For more properties about these numbers see, for example,
[8, 20 – 24]. If k = 1, we recover the Cauchy numbers c
(1)
n = cn . The Cauchy numbers cn were
introduced in [10] by the generating function
t
\mathrm{l}\mathrm{n}(1 + t)
=
\infty \sum
n=0
cn
tn
n!
.
A generalization of the above sequences was done recently in [21], using the kth q-polylogarithm
function and the Jackson’s integral. In particular, the q-poly-Bernoulli numbers are defined by
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 469
Lik,q(1 - e - t)
1 - e - t
=
\infty \sum
n=0
B(k)
n,q
tn
n!
, k \in \BbbZ , n \geq 0, 0 \leq q < 1,
where
Lik,q(t) =
\infty \sum
n=1
tn
[n]kq
is the kth q-polylogarithm function (cf. [29]), and [n]q =
1 - qn
1 - q
is the q-integer (cf. [39]). Note
that \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1[x]q = x, \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1B
(k)
n,q = B
(k)
n and \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1 Lik,q(x) = Lik(x).
The q-poly-Cauchy numbers of the first kind c
(k)
n,q are defined by using the Jackson’s q-integral
(cf. [1])
c(k)n,q =
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
(t1 . . . tk)ndqt1 . . . dqtk,
where
x\int
0
f(t)dqt = (1 - q)x
\infty \sum
n=0
f(qnx)qn.
Moreover, its exponential generating function is
Lifk,q(\mathrm{l}\mathrm{n}(1 + t)) =
\infty \sum
n=0
c(k)n,q
tn
n!
, k \in \BbbZ ,
where
Lifk,q(t) =
\infty \sum
n=0
tn
n![n+ 1]kq
(2)
is the kth q-polylogarithm factorial function (cf. [18, 21]). Note that \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1 c
(k)
n,q = c
(k)
n and
\mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1 Lifk,q(t) = Lifk(t).
In this paper, we introduce a (p, q)-analogue of the poly-Euler polynomials by
2Lik,p,q(1 - e - t)
1 + et
ext =
\infty \sum
n=0
E(k)
n,p,q(x)
tn
n!
, k \in \BbbZ , (3)
with p and q real numbers such that 0 < q < p \leq 1, and
Lik,p,q(t) =
\infty \sum
n=1
tn
[n]kp,q
is an extension of the q-polylogarithm function and we call it the (p, q)-polylogarithm function.
The polynomials E
(k)
n,p,q(0) := E
(k)
n,p,q are called (p, q)-poly-Euler numbers. The polynomial [n]p,q =
=
pn - qn
p - q
is the n-th (p, q)-integer (cf. [13, 14, 37]), it was introduced in the context of set partition
statistics (cf. [40]). Note that \mathrm{l}\mathrm{i}\mathrm{m}p\rightarrow 1[n]p,q = [n]q and \mathrm{l}\mathrm{i}\mathrm{m}p\rightarrow 1 Lifk,p,q(t) = Lifk,q(t).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
470 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
As we already mentioned the (p, q)-analogues are an extension of the q-analogues, and coincide
in the limit when p tends to 1. The (p, q)-calculus was studied in [9], in connection with quantum
mechanics. Properties of the (p, q)-analogues of the binomial coefficients were studied in [11].
The (p, q)-analogues of hypergeometric series, special functions, Stirling numbers and their gene-
ralizations, Hermite polynomials, Volkenborn integration have been studied before, see, for instance,
[2, 14, 30, 31, 33, 34, 36, 38].
The paper is divided in two parts. In Section 2, we show several combinatorial identities of
the (p, q)-poly-Euler polynomials. Some of them involving the classical Euler polynomials and
another special numbers and polynomials such as the Stirling numbers of the second kind, Bernoulli
polynomials of order s, etc. In Section 3, we introduce the (p, q)-poly-Bernoulli polynomials and
(p, q)-poly-Cauchy polynomials of both kinds, and we generalize some well-known identities of the
classical Bernoulli and Cauchy numbers and polynomials.
2. Some identities of the poly-Euler polynomials. In this section, we give several identities
of the (p, q)-poly-Euler polynomials. In particular, Theorem 2 shows a relation between the (p, q)-
poly-Euler polynomials and the classical Euler polynomials.
It is possible to give the first values of the (p, q)-polylogarithm function for k \leq 0. For example,
Li0,p,q(x) =
x
1 - x
,
Li - 1,p,q(x) =
x
(1 - px)(1 - qx)
,
Li - 2,p,q(x) =
x(1 + pqx)\bigl(
1 - p2x
\bigr) \bigl(
1 - q2x
\bigr)
(1 - pqx)
,
Li - 3,p,q(x) =
x
\bigl(
p3q3x2 + 2p2qx+ 2pq2x+ 1
\bigr)
(1 - p3x) (1 - q3x) (1 - p2qx) (1 - pq2x)
.
In general, the (p, q)-polylogarithm function for k \leq 0 is a rational function. Indeed, let k be a
nonnegative integer then
Li - k,p,q(x) =
\infty \sum
n=1
xn
[n] - k
p,q
=
\infty \sum
n=1
[n]kp,qx
n =
\infty \sum
n=1
\biggl(
pn - qn
p - q
\biggr) k
xn =
=
1
(p - q)k
\infty \sum
n=1
k\sum
l=0
\biggl(
k
l
\biggr)
pnl( - qn)k - lxn =
1
(p - q)k
k\sum
l=0
( - 1)k - l
\biggl(
k
l
\biggr)
plqk - lx
1 - plqk - lx
.
Note that from (3) we obtain that
\bigl\{
E
(k)
n,p,q(x)
\bigr\}
n\geq 0
is an Appel sequence [35]. Therefore, we have
the following basic relations.
Theorem 1. If n \geq 0 and k \in \BbbZ , then
(i) E(k)
n,p,q(x) =
n\sum
i=0
\biggl(
n
i
\biggr)
E
(k)
i,p,qx
n - i,
(ii) E(k)
n,p,q(x+ y) =
n\sum
i=0
\biggl(
n
i
\biggr)
E
(k)
i,p,q(x)y
n - i,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 471
(iii) E(k)
n,p,q(mx) =
n\sum
i=0
\biggl(
n
i
\biggr)
E
(k)
i,p,q(x)(m - 1)n - ixn - i, m \geq 1,
(iv) E(k)
n,p,q(x+ 1) - E(k)
n,p,q(x) =
n - 1\sum
i=0
\biggl(
n
i
\biggr)
E
(k)
i,p,q(x).
Theorem 2. If n \geq 1, we have
E(k)
n,p,q(x) =
\infty \sum
\ell =0
1
[\ell + 1]kp,q
\ell +1\sum
j=0
\biggl(
\ell + 1
j
\biggr)
( - 1)jEn(x - j).
Proof. From (1) and (3), we get
2Lik,p,q(1 - e - t)
1 + et
ext =
\infty \sum
\ell =0
(1 - e - t)\ell +1
[\ell + 1]kp,q
2ext
1 + et
=
=
\infty \sum
\ell =0
1
[\ell + 1]kp,q
\ell +1\sum
j=0
\biggl(
\ell + 1
j
\biggr)
( - 1)j
2e(x - j)t
1 + et
=
=
\infty \sum
\ell =0
1
[\ell + 1]kp,q
\ell +1\sum
j=0
\biggl(
\ell + 1
j
\biggr)
( - 1)j
\infty \sum
n=0
En(x - j)
tn
n!
.
Comparing the coefficients on both sides, we get the desired result.
Theorem 2 is proved.
Theorem 3. If n \geq 1, we have
E(k)
n,p,q(x) =
\infty \sum
\ell =0
\ell \sum
i=0
i+1\sum
j=0
2( - 1)\ell - i - j
[i+ 1]kp,q
\biggl(
i+ 1
j
\biggr)
(\ell - i - j + x)n.
Proof. By using the binomial series, we get
2Lik,p,q(1 - e - t)
1 + et
ext = 2
\Biggl( \infty \sum
\ell =0
( - 1)\ell e\ell t
\Biggr) \Biggl( \infty \sum
\ell =0
(1 - e - t)\ell +1
[\ell + 1]kp,q
\Biggr)
ext =
= 2
\infty \sum
\ell =0
\ell \sum
i=0
( - 1)\ell - ie(\ell - i)t
[i+ 1]kp,q
(1 - e - t)i+1ext =
=
\Biggl(
2
\infty \sum
\ell =0
\ell \sum
i=0
( - 1)\ell - ie(\ell - i)t
[i+ 1]kp,q
\Biggr) \left( i+1\sum
j=0
\biggl(
i+ 1
j
\biggr)
( - 1)je - tjext
\right) =
= 2
\infty \sum
\ell =0
\ell \sum
i=0
i+1\sum
j=0
( - 1)\ell - i+je(\ell - i - j+x)t
[i+ 1]kp,q
\biggl(
i+ 1
j
\biggr)
=
= 2
\infty \sum
\ell =0
\ell \sum
i=0
i+1\sum
j=0
( - 1)\ell - i+j
[i+ 1]kp,q
\biggl(
i+ 1
j
\biggr) \infty \sum
n=0
(\ell - i - j + x)n
tn
n!
=
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
472 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
=
\infty \sum
n=0
\infty \sum
\ell =0
\ell \sum
i=0
i+1\sum
j=0
2( - 1)\ell - i+j
[i+ 1]kp,q
\biggl(
i+ 1
j
\biggr)
(\ell - i - j + x)n
tn
n!
.
Comparing the coefficients on both sides, we get the desired result.
Theorem 3 is proved.
2.1. Some relations with other special polynomials. Jolany et al. [15] discovered several
combinatorics identities involving generalized poly-Euler polynomials in terms of Stirling numbers of
the second kind S2(n, k), rising factorial functions (x)(m), falling factorial functions (x)m, Bernoulli
polynomials \frakB
(s)
n (x) of order s, and Frobenius – Euler functions H
(s)
n (x;u). We will give similar
expressions in terms of (p, q)-poly-Euler polynomials.
Remember that the Stirling numbers of the second kind are defined by
(ex - 1)m
m!
=
\infty \sum
n=m
S2(n,m)
xn
n!
. (4)
Theorem 4. We have the following identity:
E(k)
n,p,q(x) =
\infty \sum
\ell =0
n\sum
i=\ell
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q( - \ell )(x)(\ell ), (5)
where
(x)(m) = x(x+ 1) . . . (x+m - 1), m \geq 1, with (x)(0) = 1.
Proof. From (3), (4), and by the binomial series
1
(1 - x)c
=
\infty \sum
n=0
\biggl(
c+ n - 1
n
\biggr)
xn,
we get
2Lik,p,q(1 - e - t)
1 + et
ext =
2Lik,p,q(1 - e - t)
1 + et
(1 - (1 - e - t)) - x =
=
2Lik,p,q(1 - e - t)
1 + et
\infty \sum
\ell =0
\biggl(
x+ \ell - 1
\ell
\biggr)
(1 - e - t)\ell =
=
\infty \sum
\ell =0
(x)(\ell )
\ell !
(1 - e - t)\ell
2Lik,p,q(1 - e - t)
1 + et
=
=
\infty \sum
\ell =0
(x)(\ell )
(et - 1)\ell
\ell !
\biggl(
2Lik,p,q(1 - e - t)
1 + et
e - t\ell
\biggr)
=
=
\infty \sum
\ell =0
(x)(\ell )
\Biggl( \infty \sum
n=0
S2(n, \ell )
tn
n!
\Biggr) \Biggl( \infty \sum
n=0
E(k)
n,p,q( - \ell )
tn
n!
\Biggr)
=
=
\infty \sum
\ell =0
(x)(\ell )
\infty \sum
n=0
\Biggl(
n\sum
i=0
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q( - \ell )
\Biggr)
tn
n!
=
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 473
=
\infty \sum
n=0
\Biggl( \infty \sum
\ell =0
n\sum
i=\ell
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q( - \ell )(x)(\ell )
\Biggr)
tn
n!
.
Comparing the coefficients on both sides, we have (5). Note that we use the following relation:\biggl(
x+ \ell - 1
s
\biggr)
=
(x)(\ell )
s!
.
Theorem 4 is proved.
Theorem 5. We have the following identity:
E(k)
n,p,q(x) =
\infty \sum
\ell =0
n\sum
i=\ell
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q(x)\ell , (6)
where
(x)m = x(x - 1) . . . (x - m+ 1), m \geq 1, with (x)0 = 1.
Proof. From (3) and (4), we obtain
2Lik,p,q(1 - e - t)
1 + et
ext =
2Lik,p,q(1 - e - t)
1 + et
((et - 1) + 1)x =
=
2Lik,p,q(1 - e - t)
1 + et
\infty \sum
\ell =0
\biggl(
x
\ell
\biggr)
(et - 1)\ell =
=
\infty \sum
\ell =0
(x)\ell
\ell !
(et - 1)\ell
2Lik,p,q(1 - e - t)
1 + et
=
=
\infty \sum
\ell =0
(x)\ell
\Biggl( \infty \sum
n=0
S2(n, \ell )
tn
n!
\Biggr) \Biggl( \infty \sum
n=0
E(k)
n,p,q
tn
n!
\Biggr)
=
=
\infty \sum
\ell =0
(x)\ell
\infty \sum
n=0
\Biggl(
n\sum
i=0
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q
\Biggr)
tn
n!
=
=
\infty \sum
n=0
\Biggl( \infty \sum
\ell =0
n\sum
i=\ell
\biggl(
n
i
\biggr)
S2(i, \ell )E
(k)
n - i,p,q(x)\ell
\Biggr)
tn
n!
.
Comparing the coefficients on both sides, we have (6). Note that we use the following relation:\biggl(
x
s
\biggr)
=
(x)s
s!
.
Theorem 5 is proved.
The Bernoulli polynomials \frakB
(s)
n (x) of order s are defined by\biggl(
t
et - 1
\biggr) s
ext =
\infty \sum
n=0
\frakB (s)
n (x)
tn
n!
. (7)
It is clear that if s = 1 we recover the classical Bernoulli polynomials. For some explicit formulae
of these polynomials see, for example, [28].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
474 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
Theorem 6. We have the following identity:
E(k)
n,p,q(x) =
n\sum
\ell =0
\biggl(
n
\ell
\biggr)
S2(\ell + s, s)
n - \ell \sum
i=0
\biggl(
n - \ell
i
\biggr)
\biggl(
\ell + s
s
\biggr) \frakB
(s)
i (x)E
(k)
n - \ell - i,p,q. (8)
Proof. From (3) and (7), we get
2Lik,p,q(1 - e - t)
1 + et
ext =
(et - 1)s
s!
tsext
(et - 1)s
\Biggl( \infty \sum
n=0
E(k)
n,p,q
tn
n!
\Biggr)
s!
ts
=
=
\Biggl( \infty \sum
n=0
S2(n+ s, s)
tn+s
(n+ s)!
\Biggr) \Biggl( \infty \sum
n=0
\frakB (s)
n (x)
tn
n!
\Biggr) \Biggl( \infty \sum
n=0
E(k)
n,p,q
tn
n!
\Biggr)
s!
ts
=
=
\Biggl( \infty \sum
n=0
S2(n+ s, s)
tn+s
(n+ s)!
\Biggr) \infty \sum
n=0
\Biggl(
n\sum
i=0
\biggl(
n
i
\biggr)
\frakB
(s)
i (x)E
(k)
n - i,p,q
\Biggr)
tn
n!
s!
ts
=
=
\infty \sum
n=0
\Biggl(
n\sum
\ell =0
S2(\ell + s, s)
t\ell +s
(\ell + s)!
n - \ell \sum
i=0
\biggl(
n - \ell
i
\biggr)
\frakB
(s)
i (x)E
(k)
n - \ell - i,p,q
tn - \ell
(n - \ell )!
\Biggr)
s!
ts
=
=
\infty \sum
n=0
\left( n\sum
\ell =0
\biggl(
n
\ell
\biggr)
S2(\ell + s, s)
n - \ell \sum
i=0
\biggl(
n - \ell
i
\biggr)
\biggl(
\ell + s
s
\biggr) \frakB
(s)
i (x)E
(k)
n - \ell - i,p,q
\right) tn
n!
.
Comparing the coefficients on both sides, we obtain (8).
Theorem 6 is proved.
The Frobenius – Euler functions H
(s)
n (x;u) are defined by\biggl(
1 - u
et - u
\biggr) s
ext =
\infty \sum
n=0
H(s)
n (x;u)
tn
n!
. (9)
Theorem 7. We have the following identity:
E(k)
n,p,q(x) =
n\sum
\ell =0
\biggl(
n
\ell
\biggr)
(1 - u)s
s\sum
i=0
\biggl(
s
i
\biggr)
( - u)s - iH
(s)
\ell (x;u)E
(k)
n - \ell ,p,q(i). (10)
Proof. From (3) and (9), we get
2Lik,p,q(1 - e - t)
1 + et
ext =
(1 - u)s
(et - u)s
ext
(et - u)s
(1 - u)s
2Lik,p,q(1 - e - t)
1 + et
=
=
1
(1 - u)s
\Biggl( \infty \sum
n=0
H(s)
n (x;u)
tn
n!
\Biggr)
s\sum
i=0
\biggl(
s
i
\biggr)
eti( - u)s - i 2Lik,p,q(1 - e - t)
1 + et
=
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A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 475
=
1
(1 - u)s
\Biggl( \infty \sum
n=0
H(s)
n (x;u)
tn
n!
\Biggr)
s\sum
i=0
\biggl(
s
i
\biggr)
( - u)s - i
\infty \sum
n=0
E(k)
n,p,q(i)
tn
n!
=
=
1
(1 - u)s
s\sum
i=0
\biggl(
s
i
\biggr)
( - u)s - i
\Biggl( \infty \sum
n=0
H(s)
n (x;u)
tn
n!
\Biggr) \Biggl( \infty \sum
n=0
E(k)
n,p,q(i)
tn
n!
\Biggr)
=
=
1
(1 - u)s
s\sum
i=0
\biggl(
s
i
\biggr)
( - u)s - i
\infty \sum
n=0
\Biggl(
n\sum
\ell =0
\biggl(
n
\ell
\biggr)
H
(s)
\ell (x;u)E
(k)
n - \ell ,p,q(i)
\Biggr)
tn
n!
=
=
\infty \sum
n=0
\Biggl(
1
(1 - u)s
n\sum
\ell =0
\biggl(
n
\ell
\biggr) s\sum
i=0
\biggl(
s
i
\biggr)
( - u)s - iH
(s)
\ell (x;u)E
(k)
n - \ell ,p,q(i)
\Biggr)
tn
n!
.
Comparing the coefficients on both sides, we have (10).
Theorem 7 is proved.
3. The (\bfitp , \bfitq )-poly-Bernoulli polynomials and the (\bfitp , \bfitq )-poly-Cauchy polynomials. In this
section, we introduce the (p, q)-poly-Bernoulli polynomials by means of the (p, q)-polylogarithm
function and the (p, q)-poly-Cauchy polynomials by using the (p, q)-integral. In general it is not
difficult to extend the results of [21].
The (p, q)-derivative of the function f is defined by (cf. [5, 13])
Dp,qf(x) =
\left\{
f(px) - f(qx)
(p - q)x
, if x \not = 0,
f \prime (0), if x = 0.
In particular, if p \rightarrow 1 we obtain the q-derivative [1]. The (p, q)-integral of the function f is defined
by
x\int
0
f(t)dp,qt =
\left\{
(q - p)x
\sum \infty
n=0
pn
qn+1
f
\biggl(
pn
qn+1
x
\biggr)
, if | p/q| < 1,
(p - q)x
\sum \infty
n=0
qn
pn+1
f
\biggl(
qn
pn+1
x
\biggr)
, if | p/q| > 1.
For example,
\ell \int
0
t\ell dp,qt =
1
[\ell + 1]p,q
.
We introduce the (p, q)-poly-Bernoulli polynomials by
Lik,p,q(1 - e - t)
1 - e - t
e - xt =
\infty \sum
n=0
B(k)
n,p,q(x)
tn
n!
, k \in \BbbZ .
In particular, \mathrm{l}\mathrm{i}\mathrm{m}p\rightarrow 1B
(k)
n,p,q(x) = B
(k)
n,q(x), which are the q-poly-Bernoulli polynomials studied re-
cently in [21].
The following theorem related the (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Euler poly-
nomials.
Theorem 8. If n \geq 1, we have
E(k)
n,p,q(x) + E(k)
n,p,q(x+ 1) = 2B(k)
n,p,q( - x) - 2B(k)
n,p,q(1 - x).
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476 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
Proof. From the equality
2Lik,p,q(1 - e - t)
1 + et
(1 + et)ext =
2Lik,p,q(1 - e - t)
1 - e - t
(1 - e - t)ext,
we obtain
\infty \sum
n=0
E(k)
n,p,q(x)
tn
n!
+
\infty \sum
n=0
E(k)
n,p,q(x+ 1)
tn
n!
= 2
\infty \sum
n=0
B(k)
n,p,q( - x)
tn
n!
- 2
\infty \sum
n=0
B(k)
n,p,q(1 - x)
tn
n!
.
Comparing the coefficients on both sides, we get the desired result.
Theorem 8 is proved.
The weighted Stirling numbers of the second kind, S2(n,m, x), were defined by Carlitz [6] as
follows:
ext(et - 1)m
m!
=
\infty \sum
n=m
S2(n,m, x)
tn
n!
.
Theorem 9. If n \geq 1, we have
B(k)
n,p,q(x) =
n\sum
m=0
( - 1)m+nm!
[m+ 1]kp,q
S2(n,m, x).
Proof. We obtain
\infty \sum
n=0
B(k)
n,p,q(x)
tn
n!
=
Lip,q(1 - e - t)
1 - e - t
e - xt =
\infty \sum
m=0
(1 - e - t)m
[m+ 1]kp,q
e - xt =
=
\infty \sum
m=0
( - 1)mm!
[m+ 1]kp,q
(e - t - 1)m
m!
e - xt =
\infty \sum
m=0
( - 1)mm!
[m+ 1]kp,q
\infty \sum
n=m
S2(n,m, x)
( - t)n
n!
=
=
\infty \sum
n=0
\Biggl( \infty \sum
m=0
( - 1)m+nm!
[m+ 1]kp,q
S2(n,m, x)
\Biggr)
tn
n!
.
Comparing the coefficients on both sides, we get the desired result.
Theorem 9 is proved.
The (p, q)-poly-Cauchy polynomials of the first kind are defined by
C(k)
n,p,q(x) =
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
(t1 . . . tk - x)ndp,qt1 . . . dp,qtk. (11)
Note that \mathrm{l}\mathrm{i}\mathrm{m}p\rightarrow 1C
(k)
n,p,q(x) = C
(k)
n,q(x), i.e., we obtain the q-poly-Cauchy polynomials [18, 21].
Remember that the (unsigned) Stirling numbers of the first kind are defined by
(\mathrm{l}\mathrm{n}(1 + x))m
m!
=
\infty \sum
n=m
( - 1)n - mS1(n,m)
xn
n!
. (12)
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A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 477
Moreover, they satisfy (cf. [10])
x(n) = x(x+ 1) . . . (x+ n - 1) =
n\sum
m=0
S1(n,m)xm. (13)
The weighted Stirling numbers of the first kind, S1(n,m, x), are defined by [6]
(1 - t) - x( - \mathrm{l}\mathrm{n}(1 - t))m
m!
=
\infty \sum
n=m
S1(n,m, x)
tn
n!
.
Theorem 10. If n \geq 1, we have
C(k)
n,p,q(x) =
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
= (14)
=
n\sum
m=0
S1 (n,m, x)
( - 1)n - m
[m+ 1]kp,q
. (15)
Proof. By (11), (13) and (x)n = ( - 1)n( - x)(n), we obtain
C(k)
n,p,q(x) =
n\sum
m=0
( - 1)n - mS1(n,m)
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
(t1 . . . tk - x)mdp,qt1 . . . dp,qtk =
=
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)m - \ell
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
t\ell 1 . . . t
\ell
kdp,qt1 . . . dp,qtk =
=
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)m - \ell
[\ell + 1]kp,q
=
=
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
.
Comparing the coefficients on both sides, we get (14). Finally, from the relation [6] (Eq. (5.2))
S1(n,m, x) =
n\sum
i=0
\biggl(
m+ i
i
\biggr)
xiS1(n,m+ i),
we have
C(k)
n,p,q(x) =
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
=
n\sum
\ell =0
n\sum
m=\ell
( - 1)n - mS1(n,m)
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
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478 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
=
n\sum
\ell =0
n+\ell \sum
m=\ell
( - 1)n - mS1(n,m)
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
=
n\sum
\ell =0
n\sum
m=0
( - 1)n - m+\ell S1(n,m+ \ell )
\biggl(
m+ \ell
\ell
\biggr)
( - x)\ell
[m+ 1]kp,q
=
=
n\sum
m=0
( - 1)n - m
[m+ 1]kp,q
m\sum
\ell =0
\biggl(
m+ \ell
\ell
\biggr)
S1(n,m+ \ell )x\ell =
n\sum
m=0
( - 1)n - m
[m+ 1]kp,q
S1(n,m, x).
Theorem 10 is proved.
It is not difficult to give a (p, q)-analogue of (2).
Theorem 11. The exponential generating function of the (p, q)-poly-Cauchy polynomials C(k)
n,p,q(x)
is
Lifk,p,q (\mathrm{l}\mathrm{n}(1 + t))
(1 + t)x
=
\infty \sum
n=0
C(k)
n,p,q(x)
tn
n!
, (16)
where
Lifk,p,q(t) =
\infty \sum
n=0
tn
n![n+ 1]kp,q
is the kth (p, q)-polylogarithm factorial function.
Proof. From Theorem 10, we have
\infty \sum
n=0
C(k)
n,p,q(x)
tn
n!
=
\infty \sum
n=0
n\sum
m=0
( - 1)n - mS1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
tn
n!
=
=
\infty \sum
m=0
\infty \sum
n=m
( - 1)n - mS1(n,m)
tn
n!
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
=
\infty \sum
m=0
(\mathrm{l}\mathrm{n}(1 + t))m
m!
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
=
\infty \sum
\ell =0
( - x)\ell
\ell !
\infty \sum
m=\ell
(\mathrm{l}\mathrm{n}(1 + t))m
(m - \ell )![m - \ell + 1]kp,q
=
\infty \sum
\ell =0
( - x)\ell
\ell !
\infty \sum
n=0
(\mathrm{l}\mathrm{n}(1 + t))n+\ell
n![n+ 1]kp,q
=
=
1
(1 + t)x
\infty \sum
n=0
(\mathrm{l}\mathrm{n}(1 + t))n
n![n+ 1]kp,q
=
Lifk,p,q (\mathrm{l}\mathrm{n}(1 + t))
(1 + t)x
.
Theorem 11 is proved.
Similarly, we can defined the (p, q)-poly-Cauchy polynomials of the second kind by
\widehat C(k)
n,p,q(x) =
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
( - t1 . . . tk + x)ndp,qt1 . . . dp,qtk.
We can find analogous expressions to (14) – (16).
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A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 479
Theorem 12. If n \geq 1, we have
\widehat C(k)
n,p,q(x) = ( - 1)n
n\sum
m=0
S1(n,m)
m\sum
\ell =0
\biggl(
m
\ell
\biggr)
( - x)\ell
[m - \ell + 1]kp,q
=
= ( - 1)n
n\sum
m=0
S1 (n,m, - x)
1
[m+ 1]kp,q
.
Moreover, the exponential generating function of the (p, q)-poly-Cauchy polynomials \widehat C(k)
n,p,q(x) is
(1 + t)xLifk,p,q ( - \mathrm{l}\mathrm{n}(1 + t)) =
\infty \sum
n=0
\widehat C(k)
n,p,q(x)
tn
n!
.
3.1. Some relations between (\bfitp , \bfitq )-poly-Bernoulli polynomials and (\bfitp , \bfitq )-poly-Cauchy poly-
nomials. The weighted Stirling numbers satisfy the following orthogonality relation [6]:
n\sum
\ell =m
( - 1)n - \ell S2(n, \ell , x)S1(\ell ,m, x) =
n\sum
\ell =m
( - 1)\ell - mS1(n, \ell , x)S2(\ell ,m, x) = \delta m,n,
where \delta m,n = 1 if m = n and 0 otherwise. From above relations we obtain the inverse relation
fn =
n\sum
m=0
( - 1)n - mS1(n,m, x)gm \Leftarrow \Rightarrow gn =
n\sum
m=0
S2(n,m, x)fm.
Theorem 13. The (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials of both
kinds satisfy the following relations:
n\sum
m=0
S1(n,m, x)B(k)
m,p,q(x) =
n!
[n+ 1]kp,q
, (17)
n\sum
m=0
S2(n,m, x)C(k)
m,p,q(x) =
1
[n+ 1]kp,q
, (18)
n\sum
m=0
S2(n,m, - x) \widehat C(k)
m,p,q(x) =
( - 1)n
[n+ 1]kp,q
. (19)
Proof. From Theorem 9 and the inverse relation for the weighted Stirling numbers with
fm =
( - 1)mm!
[m+ 1]kp,q
, gn = ( - 1)nB(k)
n,p,q(x),
we obtain the identity (17). The remaining relations can be verified in a similar way by using
Theorems 10 and 12.
Theorem 13 is proved.
Note that if p \rightarrow 1 we obtain Theorem 6 in [21].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
480 T. KOMATSU, J. L. RAMÍREZ, V. F. SIRVENT
Theorem 14. The (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials of both
kinds satisfy the following relations:
B(k)
n,p,q(x) =
n\sum
\ell =0
n\sum
m=0
( - 1)n - mm!S2(n,m, x)S2(m, \ell , y)C
(k)
\ell ,p,q(y), (20)
B(k)
n,p,q(x) =
n\sum
\ell =0
n\sum
m=0
( - 1)nm!S2(n,m, x)S2(m, \ell , - y) \widehat C(k)
\ell ,p,q(y), (21)
C(k)
n,p,q(x) =
n\sum
\ell =0
n\sum
m=0
( - 1)n - m
m!
S1(n,m, x)S1(m, \ell , y)B
(k)
\ell ,p,q(y), (22)
\widehat C(k)
n,p,q(x) =
n\sum
\ell =0
n\sum
m=0
( - 1)n
m!
S1(n,m, - x)S1(m, \ell , y)B
(k)
\ell ,p,q(y). (23)
Proof. We only show the proof of (22). The proofs of the remaining identities are similar. From
equations (15) and (17), we have
n\sum
\ell =0
n\sum
m=0
( - 1)n - m
m!
S1(n,m, x)S1(m, \ell , y)B
(k)
\ell ,p,q(y) =
=
n\sum
m=0
( - 1)n - m
m!
S1(n,m, x)
m\sum
\ell =0
S1(m, \ell , y)B
(k)
\ell ,p,q(y) =
=
n\sum
m=0
( - 1)n - m
m!
S1(n,m, x)
m!
[m+ 1]kp,q
= C(k)
n,p,q(x).
Theorem 14 is proved.
Finally, we show some relations between (p, q)-poly-Cauchy polynomials of both kinds.
Theorem 15. If n \geq 1, we have
( - 1)n
C
(k)
n,p,q(x)
n!
=
n\sum
m=1
\biggl(
n - 1
m - 1
\biggr) \widehat C(k)
m,p,q(x)
m!
, (24)
( - 1)n
\widehat C(k)
n,p,q(x)
n!
=
n\sum
m=1
\biggl(
n - 1
m - 1
\biggr)
C
(k)
m,p,q(x)
m!
. (25)
Proof. From definition of the (p, q)-poly-Cauchy polynomials of the first kind, we get
( - 1)n
C
(k)
n,p,q(x)
n!
= ( - 1)n
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
(t1 . . . tk - x)n
n!
dp,qt1 . . . dp,qtk =
= ( - 1)n
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
\biggl(
t1 . . . tk - x
n
\biggr)
dp,qt1 . . . dp,qtk =
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A (p, q)-ANALOGUE OF POLY-EULER POLYNOMIALS . . . 481
=
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
\biggl(
x - t1 . . . tk + n - 1
n
\biggr)
dp,qt1 . . . dp,qtk.
By using the Vandermonde convolution
n\sum
k=0
\biggl(
r
k
\biggr) \biggl(
s
n - k
\biggr)
=
\biggl(
r + s
n
\biggr)
with r = x - t1 . . . tk and s = n - 1, we obtain
( - 1)n
C
(k)
n,p,q(x)
n!
=
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
n\sum
\ell =0
\biggl(
x - t1 . . . tk
\ell
\biggr) \biggl(
n - 1
n - \ell
\biggr)
dp,qt1 . . . dp,qtk =
=
n\sum
\ell =0
\biggl(
n - 1
n - \ell
\biggr) 1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
\biggl(
x - t1 . . . tk
\ell
\biggr)
dp,qt1 . . . dp,qtk =
=
n\sum
\ell =0
\biggl(
n - 1
n - \ell
\biggr)
1
\ell !
1\int
0
. . .
1\int
0\underbrace{} \underbrace{}
k
( - t1 . . . tk + x)\ell dp,qt1 . . . dp,qtk =
n\sum
\ell =0
\biggl(
n - 1
n - \ell
\biggr) \widehat C(k)
\ell ,p,q(x)
\ell !
.
The proof of (25) is similar.
Theorem 15 is proved.
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Received 08.02.17,
after revision — 17.05.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
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| id | umjimathkievua-article-6048 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:51Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/27/ed21b3255c5924f92c4f1450e62fec27.pdf |
| spelling | umjimathkievua-article-60482022-03-26T11:01:37Z A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. Поліном Ейлера Poly-Euler Polynomial UDC 517.5 We introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. &nbsp;These new sequences are generalizations of the poly-Euler numbers and polynomials. &nbsp;We give several combinatorial identities and properties of these new polynomials, and also show some relations with $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials.&nbsp;The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue. UDC 517.5 Введено $(p,q)$-аналоги поліейлерівських поліномів і чисел за допомогою $(p,q)$-полілогарифмічної функції, які є узагальненнями поліейлерівських поліномів і чисел. &nbsp;Знайдено властивості цих поліномів і наведено деякі відповідні комбінаторні рівності. &nbsp;Також показано зв'язок із $(p,q)$-поліномами типу Бернуллі та Коші. &nbsp;Ці $(p,q)$-аналоги узагальнюють відому концепцію $q$-аналогів. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6048 10.37863/umzh.v72i4.6048 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 4 (2020); 467-482 Український математичний журнал; Том 72 № 4 (2020); 467-482 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6048/8701 |
| spellingShingle | Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. Komatsu, T. Ramírez, J. L. Sirvent, V. F. A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_alt | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_full | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_fullStr | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_full_unstemmed | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_short | A $(p,q)$ analogue of Poly-Euler polynomials and some related polynomials |
| title_sort | $(p,q)$ analogue of poly-euler polynomials and some related polynomials |
| topic_facet | Поліном Ейлера Poly-Euler Polynomial |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6048 |
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