Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

: The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and ﬁnite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.


Introduction
Special polynomials and their generating functions have important roles in many branches of mathematics, probability, statistics, mathematical physics and also engineering. Since polynomials are suitable for applying well-known operations such as derivative and integral, polynomials are very useful to study real-world problems in aforementioned areas. For instance, generating functions for special polynomials with their congruence properties, recurrence relations, computational formulae and symmetric sum involving these polynomials has been studied by many authors in recent years (cf. ).
In this article, by combining the Euler's formula with generating functions for two parametric kinds of Eulerian-type polynomials, their functional equations and partial derivative equations, we give many formulae and relations including the Stirling numbers, Fubini-type polynomials, two parametric kinds of Eulerian-type polynomials, and Apostol-type numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials, the Apostol-Euler numbers and polynomials, and the Apostol-Genocchi numbers and polynomials.
Throughout this article, we use the following notations and definitions: Let N = {1, 2, 3, ...}, N 0 = N ∪ {0}, Z denote the set of integers, R denote the set of real numbers and C denote the set of complex numbers.
In [7], we defined the following generating function for the Fubini-type polynomials a (m) Substituting x = 0 into (11), we have a (m) where a (m) n denotes the so-called Fubini-type numbers of order m (cf. [7]). In [9], we constructed the following generating functions for two kinds of Hermite-based r-parametric Milne-Thomson-type polynomials: Let where where The rest of this article is briefly summarized as follows: In Section 2, we define generating functions for two parametric kinds of Eulerian-type polynomials. By using Euler's formula and these generating functions with their functional equations, we give relations and computation formulae for these polynomials. By using these formulae, we give a few values of these polynomials. Finally, we give some relations among the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Frobenius-Euler polynomials, the Apostol-Genocchi polynomials, the Stirling numbers, the Fubini-type polynomials and these polynomials.
In Section 3, we give functional equations and differential equations of these generating functions. By using these functional and differential equations, we derive derivative formulae and finite combinatorial sums involving the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Apostol-Genocchi numbers and for two parametric kinds of Eulerian-type polynomials. Section 4 is the conclusions section.

New Families of Two Parametric Kinds of Eulerian-Type Polynomials
In this section, we construct generating functions for two parametric kinds of Eulerian-type polynomials. By combining these functions with the Euler's formula, we give not only fundamental properties of these polynomials, but also new identities and relations related to the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Apostol-Genocchi numbers and for two parametric kinds of Eulerian-type polynomials.
We define the following generating functions for two parametric kinds of Eulerian-type polynomials: and where are so-called two parametric kinds of Eulerian-type polynomials of order k 1 and k 2 , respectively.
Note that the symbols C and S occurring in the superscripts on the right-hand sides of Equations (14) and (15) denote the trigonometric cosine and the trigonometric sine functions, respectively. (12) and (13), we have the following identities, respectively: (14) and (15), we get the following generating functions, respectively: Remark 3. In ( [19], p. 10), the second author defined following generating function for generalized Eulerian-type polynomials of order m: Substituting a = 1, b = c = e into the above equation, we have and F .
Proof. By combining Equations (14) and (15) with the Euler's formula, we obtain Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result. Theorem 2. Let n ∈ N 0 . Then, we have Proof. By using (3), (9) and (14), we get the following functional equation: Using the aforementioned equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result. Proof. By using (3), (10) and (15), we obtain the following functional equation: Using the aforementioned functional equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Proof. By using (14) and (6), we obtain the following functional equation: By using the aforementioned equation, we get Therefore, n−2j (x; λ, u) y 2j t n n! .
Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Using the aforementioned equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Combining (19) with (8), we arrive at the following corollary: Theorem 6. Let n ∈ N 0 . Then, we have Proof. Using (8), (9) and (14), we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Proof. Using (8), (10) and (15), we obtain Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
By using (20) and (21) Using the aforementioned Equation (17) and Euler's formula, we obtain Comparing the coefficients of t n n! on both sides of the aforementioned equation, we get the following theorem: n−j (x, y; λ, u) .
Using the aforementioned equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result. n−j (x, 2y; λ, u) .
From the above equation, we have Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Combining (1) with (14), we get the following functional equation: By using the aforementioned equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result. Proof.
Combining (1) with (15), we obtain the following functional equation: Using the above functional equation, observe that proof of the assertion of (25) follows precisely along the same lines as that proof of the assertion of (24), and so we omit it.
Theorem 13. Let n ∈ N 0 . Then, we have Proof. Combining (5) with (14), we have the following functional equation: Using the aforementioned equation, we get Therefore, Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Proof. By using (5) and (15), we derive the following functional equation: From the above equation, observe that proof of the assertion of (27) follows precisely along the same lines as that proof of assertion of (26), and so we omit it.
Theorem 15. Let n ∈ N 0 . Then, we have Proof. By using (2) and (14), we derive the following functional equation: Using the aforementioned equation, we get Therefore, ∞ ∑ n=0 H (C,k 1 ) n (x, y; λ, u) t n n! = u − 1 2u Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
Comparing the coefficients of t n n! on both sides of the aforementioned equation, we arrive at the desired result.
From the above equation, observe that proof of the assertion of (33) follows precisely along the same lines as that proof of the assertion of (32), and so we omit it.