Identities associated with Milne–Thomson type polynomials and special numbers

The purpose of this paper is to give identities and relations including the Milne–Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p-adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.


Introduction
n ∈ N. , and the references therein). In order to prove identities, relations, formulas, and combinatorial sums related to the special numbers and polynomials of this paper, we need the following generating functions for these special numbers and polynomials including some basic properties of them.
The Bernoulli polynomials are defined by Substituting x = 0 into (1), we have the Bernoulli numbers B n : The Euler polynomials are defined by Substituting x = 0 into (2), we have the Euler numbers E n : E n = E n (0) (cf. , and the references therein). The array polynomials are defined by where v ∈ N 0 . By (3), we have [1,3,24,26]). Substituting x = 0 into (3), we have the Stirling numbers of the second kind S 2 (n, v): which defined by means of the following generating function: where v ∈ N 0 (cf. [3,5,7,24,30,31] and the references therein).

p-adic integral
Here, we survey some properties of the p-adic integral. Thus, we give some notations and definitions. Z p denotes the set of p-adic integers, Q p denotes the set of p-adic rational numbers, K denotes a field with a complete valuation and C p is completion of the algebraic closure of Q p . C 1 (Z p → K) denotes the set of continuous derivative functions. Let f (x) ∈ C 1 (Z p → K). Kim [14] defined the p-adic q-integral as follows: where q ∈ C p with |1 -q| p < 1, f ∈ C 1 (Z p → K), is a q-distribution on Z p (cf. [14]).
Remark 1 If q → 1, then (11) reduces to the Volkenborn integral (or so-called the bosonic integral): where denotes the Haar distribution on Z p (cf. [23]), and the references therein).
Remark 2 If q → -1, then (11) reduces to the fermionic p-adic integral: and The Bernoulli numbers are also given by the following bosonic p-adic integral: (cf. [14,23]). On the other hand, the Euler numbers are also given by the following fermionic p-adic integral: (cf. [13]).
The Daehee numbers D n are introduced by the following bosonic p-adic integral: (cf. [8,11], [21, p. 45], and the references therein). The Changhee numbers Ch n are introduced by the following fermionic p-adic integral: (cf. [12][13][14], and the references therein). We now summarize the results of this paper as follows: In Sect. 2, by using generating functions and their functional equations, we give some identities including the three-variable polynomials y 6 (n; x, y, z; a, b, v), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.
In Sect. 3, by using p-adic integrals, we give some identities, combinatorial sums and relations related to the three-variable polynomials y 6 (n; x, y, z; a, b, v), the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind) and other special numbers such as the Daehee numbers and the Changhee numbers.
In Sect. 4, by using generating functions associated with trigonometric functions and the central factorial numbers of the second kind, we derive identities related to the central factorial numbers of the second kind, the array polynomials, and combinatorial sum.

Identities related to the Hermite polynomials, array polynomials and Stirling numbers of the second kind: generating functions and their functional equations approach
In this section, by applying generating functions and their functional approach, we derive some identities including the three-variable polynomials y 6 (n; x, y, z; a, b, v), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.
Theorem 1 where [x] denotes the greatest integer function.
Proof Substituting b = 0, h(t, v) = t v and f (t, a) = 1 a! (e t -1) a with a ∈ N 0 and z = 1 into (10), we get the following functional equation: x, y, 1; a, 0 Combining the above equation with (10), and (3), we obtain Equating the coefficients of t n n! on both sides of the equation, we arrive at the desired result.
Proof Proof of this theorem was also given in [33,Theorem 2]. We now briefly give another proof of this theorem. Substituting b = 0, h(t, v) = t v and f (t, a) = 1 a! (e t -1) a with a ∈ N 0 and z = 1 into (10), we construct the following functional equation: Combining the above equation with (10), (7), and (4), we get Therefore ∞ n=0 y 6 (n; x, y, 1; Equating the coefficients t n n! on both sides of the equation, we arrive at the desired result. Combining (18) and (19), we arrive at the following theorem.

Identities and relations related to Stirling numbers and other special numbers: p-adic integral approach
In this section, by applying p-adic integrals approach, we derive some identities, combinatorial sums and relations related to the three-variable polynomials y 6 (n; x, y, z; a, b, v), the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind) and other special numbers such as the Daehee numbers and the Changhee numbers. We assume that |e t -1| < 1. We obtain ∞ m=0 y 6 (m; 0, 0, z; 1, 1, 1) where (z) n = z(z -1) · · · (zn + 1).
Combining the above equation with (4), since S 2 (m, n) = 0 for m < n, we obtain Equating the coefficients of t m m! on both sides of the equation, we arrive at the desired result.
We observe that, using the orthogonality relation of the Stirling numbers, Eq. (23) reduces to the following well-known relations for the Bernoulli numbers: (-1) n n!S 2 (m, n) n + 1 (cf. [3,7,11,17,18,25], and the references therein). By applying the fermionic p-adic integral to (20), we obtain and we also obtain Combining the above equations, we get the following theorem.
Integrating Eq. (20) with respect to z from 0 to 1, we obtain Since where i 2 = -1, after some elementary calculations, we obtain (-1) n-m (i) n 2 n-2m S n 2m (-m) Combining (27) and (28), and also using the Cauchy product formula for a series product, we also obtain Since combining (29) and (30), after some calculations, we arrive at the following theorem including a relation between the central factorial numbers of the second kind and the array polynomials. By combining (29) and (30), we also get following corollary.
With the help of Eq. (31), we also obtain the following combinatorial sum.

Conclusion
This paper contains many kind of identities and relations related to the Milne-Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By applying not only p-adic integral, but also the Riemann integral methods, many identities relations and formulas related to the aforementioned numbers and polynomials, and also the combinatorial sums are given. By using the orthogonality relation of the Stirling numbers, explicit formulas for the Bernoulli numbers and the Euler numbers are provided. The results of this paper have potential applicability to physics, engineering and other related fields, especially branches of mathematics.