Unification of Two-Variable Family of Apostol-Type Polynomials with Applications

In this paper, the two-variable unified family of generalized Apostol-type polynomials is introduced, and some implicit forms and general symmetry identities are derived. Also, we obtain new degenerate Apostol-type numbers and polynomials constructed from the new 2-variable unified family. We derive explicit formulae of polynomials and identities that include some special numbers and polynomials. In addition, a probabilistic representation of the new family and some statistical properties are obtained.

(2) e 2-variable general polynomials p n (x, y) contain a number of important special polynomials of two variables.
Generating functions for certain members that belong to the 2VGP are given as follows: e higher-order Hermite polynomials, sometimes called the Gould-Hopper polynomials H (m) n (x, y) are dened by [2].
Khan et al. [9] introduced the 2-variable Apostol type polynomials of order α, by Araci et al. [10] introduced and investigate a new unified class of generalized Apostol type polynomials by the following generating function: In [11], Cartilz introduced a new study of degenerate versions of Bernoulli and Euler numbers and polynomials. Lately, many researchers have begun to study the degenerate Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate Hermite polynomials and numbers, and so on (see [12][13][14]).
Furthermore, mathematicians have done recently a few interesting works on Apostol-type polynomials in the field of approximation theory (for more details, see [15][16][17][18]).
is manuscript is a modified and extended version of [19]. e paper is organized as follows: in Section 2, we construct a new version of the 2-variable Apostol-type polynomials and numbers and related special polynomials and numbers. In Section 3, by using generating functions, we investigate some summation formulae, explicit expression, and some symmetric identities. In Section 4, we construct a new version of 2-variable degenerate Apostol-type polynomials and numbers and related special polynomials. In Section 5, we obtain the generating functions for new special polynomials that belong to the new version of 2-variable unified Apostol-type polynomials. In Section 6, the probabilistic representation of the new family and its statistical properties are presented.

Unification of Two-Variable Apostol-
Type Polynomials e two-variable unified family of generalized Apostol-type polynomials of order r, denoted by pU (r) n (x, y; a, b, c, ], μ; α r ) is defined as the Apostol type convolution of the 2-variable general polynomials p n (x, y). Definition 1. Let a, b and c ∈ R + and a ≠ b. A new generalization of the Apostol Hermite-Genocchi polynomials pU (r) n (x, y; a, b, c, ], μ; α r ) for nonnegative integer n is defined by the generating function where r ∈ C; α r � (α 0 , α 1 , . . . , α r−1 ) is a sequence of complex numbers. Setting c � e and φ(y, t) � 1 in (13), we get the following definition.

Definition 2. A unified family U (r)
n (x; a, b, ], μ; α r ) of generalized Apostol-type polynomials is given by Remark 1. Setting x � 0 in (14), then we obtain the new unified family of generalized Apostol-type numbers, which is defined as Also, setting c � e in (13), we get the following definition.
Definition 3. e two-variable unified family of generalized Apostol-type polynomials of order (r) pU (r) n (x, y; a, b; ], μ; α r ) is defined by the following generating function We obtain the series definition of pU (r) n (x, y; a, b, ], μ; α r ) by the following theorem.

Theorem 1. e two-variable unified family of generalized Apostol-type of order (r)pU (r)
n (x, y; a, b, ], μ; α r ) is defined by the following series: Proof. Using equation (13) and the Cauchy-product rule, we can easily yield (17). Now, by taking certain values of parameters in equations (13) and (16), we can find the generating functions and other results for the mixed special polynomials related to pU (r) n (x, , y; a, b, ], μ; α r ). We present the generating function and series definitions for these polynomials in Table 1.
n (x, y; a, b, c, μ; α r ) can be expressed as Proof. e left-hand side of (13) is equal to By (13), we get By comparing the coefficients on both sides in the last equation, we obtain (18).

Implicit Summation Formulas for the Two-Variable Unified Family of Generalized Apostol-Type Polynomials
Theorem 3. Let a, b > 0 and a ≠ b. en for x, y, z ∈ R and n ≥ 0. e following implicit summation formula for pU (r) n (x, y; a, b; μ, ]; α r ) holds true as follows: Proof. Replacement of x by x + z in (16) gives Replacing n with n − m in the right-hand side, hence equating the coefficients of t in both sides of the last equation yields (21).
Proof. Replacing x by x + z in (16) gives We get Equating the coefficients of t n on both sides, yields (23).

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e unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, [21] 9 n (x, y; a, b, c, k; α r ) e generalization of Apostol-Hermite Genocchi polynomials, [22] 4 Journal of Mathematics Theorem 5. Let a, b > 0 and a ≠ b. en for x, y, z ∈ R and n ≥ 0, the following implicit summation formula for pU (r) n (x, y; a, b; μ, ]; α r ) holds true as follows: Proof. Replacing t by t + u in the generating function (16) and using the following rule [23]: We obtain Replacing x by z in the previous equation and then equal both sides, we get Now replacing n by n − p, m by m − q, and using the Cauchy-product rule in the left-hand side of (31), we get  e following implicit summation formula for pU (r) n (x, y; a, b; μ, ]; α r ) holds true as follows:

Journal of Mathematics
Proof. Let (36) en the expression for G(t) is symmetric in c and d and we can expand G(t) into series in two ways. Firstly Secondly

Applications
e 2VGP family p n (x, y) contains a number of important special polynomials of two variables. Some members belonging to the 2VGP family are considered in Section 1. We notice that for every member belonging to the 2VGP, there is a new special polynomial that belongs to the pU (r) n (x, y, a, b, ], μ; α r ) family. us, by selecting a suitable choice for the function φ(y, t) in equation (16), the generating function for the corresponding member belongs to pU (r) n (x, y, a, b, ], μ; α r ) is a family that can be obtained.

Remark 2.
For m � 2, the H (m) n (x, y) reduce to H n (x, y). erefore, setting m � 2 in equation (39), we obtain the following generating function for the 2-variable Hermite Apostol type polynomials, denoted by HU (r) n (x, y, a, b, ], μ; α r ) as follows: e series definitions and some results for the 2-variable Hermite Apostol type polynomials HU (r) n (x, y, a, b, ], μ; α r ) can be deduced by setting m � 2 in the results given in Table 2.
Example 2. Setting φ(y, t) � C 0 (−yt m ) (for which the p n (x, y) reduce to the mL n (y, x)) in the left-hand side of generating function (16), we find that the resultant 2-variable generalized Laguerre Apostol type polynomials (2VGLATP), denoted by mL U (r) n (y, x, a, b, ], μ; α r ) are defined by the following generating function: e series definitions and other results for the 2VGLATP mL U (r) n (y, x, a, b, ], μ; α r ) are given in Table 3.

Remark 3.
Since for m � 1 and y ⟶ − y, then mL n (x, y) reduce to the L n (x, y). erefore, setting m � 1 and y ⟶ − y in equation (41), we obtain the generating function for the 2-variable Laguerre Apostol type polynomials, denoted by e series definitions and other results for the LU (r) n (y, x, a, b, ], μ; α r ) can be obtained by setting m � 1 and y ⟶ − y in the results given in Table 3.

Remark 4.
Since for x � 1, the L n (x, y) reduce to the classical Laguerre polynomials L n (y). erefore, setting x � 1 in equation (42), we obtain the following generating function for the Laguerre Apostol type polynomials, denoted by LU (r) n (y, a, b, ], μ; α r ): ∞ n�0 LU (r) n y, a, b, ], μ; α r t n n! � (−1) r t r] 2 rμ e series definitions and other results for the LU (r) n (y, a, b, ], μ; α r ) can be obtained by setting m � 1, y ⟶ − y, and x � 1 in the results given in Table 3.
Setting suitable values of the parameters in the results of the mL U (r) n (y, x, a, b, ], μ; α r ), we obtain results for 2-variable generalized Laguerre-Apostol polynomials related to mL U (r) n (y, x, a, b, ], μ; α r ) (for more details see [9]).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest with this study.