A ( p, q )-Analog of Poly-Euler Polynomials and Some Related Polynomials
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A (p, q)-ANALOG OF POLY-EULER POLYNOMIALS AND SOME RELATED POLYNOMIALS T. Komatsu,1 J. L. Ram´ırez,2,3 and V. F. Sirvent4
UDC 517.5
We introduce a (p, q)-analog 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 present 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)-analogs generalize the well-known concept of q-analog.
1. Introduction The Euler numbers are defined by the generating function 1
X tn 2 . = E n et + e−t n! n=0
The sequence (En )n counts the numbers of alternating n-permutations. An n-permutation σ is alternating if the n − 1 differences σ(i + 1) − σ(i) for i = 1, 2, . . . , n − 1 have alternating signs. Thus, (1324) and (3241) are alternating permutations (cf. [10]). The Euler polynomials are given by the generating function 1
X 2ext tn E (x) = . n et + 1 n! n=0
Note that En = 2n En (1/2). Numerous kinds of generalizations of these numbers and polynomials can be found in the literature (see, e.g., [39]). In particular, we are interested in the poly-Euler numbers and polynomials (cf. [12, 15, 16, 32]). (k) The poly-Euler polynomials En (x) are defined by the following generating function: 1
2Lik (1 − e−t ) xt X (k) tn , e = E (x) n 1 + et n! n=0
k 2 Z,
where 1 n X t Lik (t) = nk
(1)
n=1
1
Department of Mathematics, School of Sciences, Zhejiang Sci-Tech University, Hangzhou, China; e-mail: [email protected]. Universidad Nacional de Colombia, Bogot´a, Colombia; e-mail: [email protected]. 3 Corresponding author. 4 Universidad Cat´olica del Norte, Antofagasta, Chile; e-mail: [email protected]. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 467–482, April, 2020. Original article submitted February 8, 2017; revision submitted May 17, 2018. 536
0041-5995/20/7204–0536
© 2020
Springer Science+Business Media, LLC
A (p, q)-A NALOG OF P OLY-E ULER P OLYNOMIALS AND S OME R ELATED P OLYNOMIALS
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is the k th polylogarithm function. Note that if k = 1, then Li1 (t) = − log(1 − t). Hence, En(1) (x) = En−1 (x)
for n ≥ 1.
It is also possible to define the poly-Bernoulli and poly-Cauchy numbers and polynomials by using the k th (k) polylogarithm function. Thus, in particular, the poly-Bernoulli numbers Bn were introduced by Kaneko [17] by using the following generating function: 1
Lik (1 − e−t ) X (k) tn , = Bn 1 − e−t n! n=0
k 2 Z.
If k = 1, then we get Bn(1) = (−1)n Bn
for n ≥ 0,
where Bn are the Bernoulli numbers. Remember that the Bernoulli numbers Bn are defined by the generating function 1
X t tn = . B n et − 1 n! n=0
The poly-Bernoulli numbers and polynomials were studied in several papers; among other references, see [3, 4, 7, 8, 21, 22, 25–27]. (k) The poly-Cauchy numbers of the first kind cn were introduced by the first author in [19]. They are defined as follows: c(k) n =
Z1
...
Z1
(t1 . . .
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