A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials

In the present article, 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. Moreover, we show some relations with the $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue.


Introduction
The Euler numbers are defined by the generating function The sequence (E n ) n counts the numbers of alternating n-permutations. A n-permutation σ is alternating if the n − 1 differences σ(i + 1) − σ(i) for i = 1, 2, . . . , n − 1 have alternating signs. For example, (1324) and (3241) are alternating permutations (cf. [9]). The Euler polynomials are given by the generating function 2e xt e t + 1 = ∞ n=0 E n (x) t n n! . (1) Note that E n = 2 n E n (1/2). Many kinds of generalizations of these numbers and polynomials have been presented in the literature (see, e.g., [32]). In particular, we are interested in the poly-Euler numbers and polynomials (cf. [11,14,15,27]).
The poly-Euler polynomials E (k) n (x) are defined by the following generating function where Li k (t) = ∞ n=1 t n n k (2) is the k-th polylogarithm function. Note that if k = 1, then Li 1 (t) = − log(1 − t), therefore E (1) n (x) = E n−1 (x) for n ≥ 1. It is also possible to define the poly-Bernoulli and poly-Cauchy numbers and polynomials from the k-th polylogarithm function. In particular, the poly-Bernoulli numbers B (k) n were introduced by Kaneko [16] by using the following generating function If k = 1 we get B The poly-Bernoulli numbers and polynomials have been studied in several papers; among other references, see [2,3,6,7,20,21]. The poly-Cauchy numbers of the first kind c (k) n were introduced by the first author in [18]. They are defined as follows where (x) n = x(x − 1) · · · (x − n + 1)(n ≥ 1) with (x) 0 = 1. Moreover, its exponential generating function is where Lif k (t) = ∞ n=0 t n n!(n + 1) k is the k-th polylogarithm factorial function. For more properties about these numbers see for example [7,19,20,21,22,23]. If k = 1, we recover the Cauchy numbers c  c n t n n! .
A generalization of the above sequences was done recently in [20], using the k-th qpolylogarithm function and the Jackson's integral. In particular, the q-poly-Bernoulli numbers are defined by where is the k-th q-polylogarithm function (cf. [25]), and [n] q = 1−q n 1−q is the q-integer (cf. [32]).
The q-poly-Cauchy numbers of the first kind c (k) n,q are defined by using the Jackson's qintegral (cf. [1]) Moreover, its exponential generating function is is the k-th q-polylogarithm factorial function (cf. [20,17]). Note that lim q→1 c (k) n,q = c (k) n and lim q→1 Lif k,q (t) = Lif k (t). In this paper, we introduce a (p, q)-analogue of the poly-Euler polynomials by with p and q real numbers such that 0 < q < p ≤ 1, and 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 = p n −q n p−q is the n-th (p, q)-integer (cf. [12,13,30]), it was introduced in the context of set partition statistics (cf. [33]). Note that lim p→1 [n] p,q = [n] q and lim p→1 Lif k,p,q (t) = Lif k,q (t). 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 [8], in connection with quantum mechanics. Properties of the (p, q)-analogues of the binomial coefficients were studied in [10]. The (p, q)-analogues of hypergeometric series, special functions, Stirling numbers, Hermite polynomials have been studied before, see for instance [13,26,29,31].
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.

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 ≤ 0. For example, , In general, the (p, q)-polylogarithm function for k ≤ 0 is a rational function. Indeed, let k be a nonnegative integer then Proof. From (2) and (9) we get Comparing the coefficients on both sides, we get the desired result.
Proof. By using the binomial series we get Comparing the coefficients on both sides, we get the desired result. 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 Theorem 4. We have the following identity Proof. From (9) and (10), and by the binomial series Comparing the coefficients on both sides, we have (11). Note that we use the following relation Theorem 5. We have the following identity where Proof. From (9) and (10) Comparing the coefficients on both sides, we have (12). Note that we use the following relation The Bernoulli polynomials B (s) n (x) of order s are defined by It is clear that if s = 1 we recover the classical Bernoulli polynomials. For some explicit formulae of these polynomials see for example [24].
The Frobenius-Euler functions H Proof. From (9) and (15) Comparing the coefficients on both sides, we get (16).

The (p, q)-poly Bernoulli Polynomials and the (p, q)-poly 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 [20]. The (p, q)-derivative of the function f is defined by (cf. [4,12]) In particular if p → 1 we obtain the q-derivative [1]. The (p, q)-integral of the function f is defined by For example, We introduce the (p, q)-poly Bernoulli polynomials by In particular, lim p→1 B n,q (x), which are the q-poly-Bernoulli polynomials studied recently in [20]. The following theorem related the (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Euler polynomials.
Proof. From the following equality Comparing the coefficients on both sides, we get the desired result.
The weighted Stirling numbers of the second kind, S 2 (n, m, x), were defined by Carlitz [5] as follows Theorem 9. If n ≥ 1, we have Comparing the coefficients on both sides, we get the desired result.
Remember that the (unsigned) Stirling numbers of the first kind are defined by Moreover, they satisfy (cf. [9]) The weighted Stirling numbers of the first kind, S 1 (n, m, x), are defined by ( [5]) Theorem 10. If n ≥ 1, we have Proof. By (17), (19) and (x) n = (−1) n (−x) (n) , we have It is not difficult to give a (p, q)-analogue of (8).
Theorem 11. The exponential generating function of the (p, q)-poly-Cauchy polynomials C where is the k-th (p, q)-polylogarithm factorial function.
Proof. From Theorem 10 we have Similarly, we can defined the (p, q)-poly-Cauchy polynomials of the second kind by We can find analogous expressions to (20), (21) and (22).
Moreover, the exponential generating function of the (p, q)-poly-Cauchy polynomials C (k) n,p,q (x) is n,p,q (x) t n n! .
3.1. Some relations between (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials. The weighted Stirling numbers satisfy the following orthogonality relation [5]:  , and g n = (−1) n B (k) n,p,q (x), we obtain the identity (26). The remaining relations can be verified in a similar way by using Theorems 10 and 12.
Note that if p → 1 we obtain Theorem 6 in [20].
The proof of (34) is similar.