SOME NEW CLASSES OF GENERALIZED LAGRANGE-BASED APOSTOL TYPE HERMITE POLYNOMIALS

In this paper, we present a general family of Lagrange-based Apostoltype Hermite polynomials thereby unifying the Lagrange-based Apostol HermiteBernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We further define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are obtained by using different analytical means and applying generating functions.

The details of the generalized Apostol-Bernoulli polynomials B (α) n (x) of order α, the generalized Apostol-Euler polynomials E (α) n (x) of order α and the generalized Apostol-Genocchi polynomials G (α) n (x) of order α found in [2,6,[8][9][10][11][12][13][14].The polynomials B It is easy to see that n (x; 1) and G (α) n (x) = G (α) n (x; 1).Recently, Srivastava et al. [18] introduce and investigate the following class of Lagrange-based Apostol type polynomials x) are defined by means of the following generating function: In the particular cases when k = 0 and k = 1, we define the Lagrange-based Apostol-Bernoulli polynomials are defined by means of the following generating function are defined by means of the following generating function . Furthermore, we define the Lagrange-based Apostol-Euler polynomials as follows.
Definition 1.4.The Lagrange-based Apostol-Euler polynomials x) are defined by means of the following generating function: ).The 2-variable Hermite Kampé de Fériet polynomials (2VHKdFP) H n (x, y) [3,5] are defined as are solutions of the heat equation H n (x, 0) = x n .The higher order Hermite polynomials, sometimes called the Kampé de Fériet polynomials of order m or the Gould Hopper polynomials (GHP) H (m) n (x, y) defined by the generating function [1] are a solution of the generalized heat equation ) where H n (x) are the classical Hermite polynomials [1].
The Lagrange-based generalizations which we have introduced above, enable us to obtain new and useful relations between the Apostol-Bernoulli polynomials, Apostol-Euler polynomials and the Apostol-Genocchi polynomials.In this paper, we first study several elementary properties of the generalized Lagrangebased Apostol type Hermite polynomials x, y).Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

Definitions and Basic Properties of the Generalized Lagrange
In this section, we give Lagrange-based Apostol type Hermite polynomials and give the explicit representations and list basic properties of the generalized Lagrangebased Apostol-type Hermite polynomials x, y) are defined by means of the following generating function: In the particular cases, when k = 0 and k = 1, we define the Lagrangebased Apostol Hermite-Bernoulli polynomials H B x, y) are defined by means of the following generating function: x, y) are defined by means of the following generating function: . Furthermore, we define the Lagrange-based Apostol Hermite-Euler polynomials as follows.
Proof.It is fairly straightforward to observe from (1.1) and (2.2) Comparing the coefficient of t n in both sided, we get the result (2.16 and (2.22)

Summation formulae for generalized Lagrange-based Apostol-type Hermite polynomials
First, we derive the following result involving generalized Lagrange-based Apostoltype Hermite polynomials by using series rearrangement techniques.We now begin with the following theorem.
(3.1) Proof.By the exploiting generating function (2.1) and using the (1.10) (3.2) Proof.Applying the definition (2.1), we have Replacing n by n − m in the r.h.s and comparing the coefficient of t n , we get the result (3.2).
Proof.Using the definition (2.1), we have Replacing n by n − 2j in above equation, we get Comparing the coefficient of t n , we get the result (3.3).
Theorem 3.4.The following implicit summation formulae for Lagrange-based Apostol type Hermite polynomials x, y) holds true: Proof.By the definition of generalized Lagrange-based Apostol type polynomials, we have Replacing n by n − m, we have Equating their coefficients of t n leads to formula (3.4).

General symmetry identities
In this section, we establish general symmetry identities for the generalized Lagrangebased Apostol type Hermite polynomials T x) by applying the generating functions (2.1) and (1.6).Such type of works, introduced here by the approach given in the recent works of Khan [7] and Pathan and Khan [15][16][17].
Theorem 4.1.Let a, b > 0 and a = b.Then for x, y ∈ R and n ≥ 0, the following identity holds true: (4.2) Then the expression for G(t) is symmetric in a and b and we can expand G(t) into series in two ways to obtain: On the similar lines we can show that Let a, b, c > 0 and a = b.Then for x, y ∈ R and n ≥ 0, the following identity holds true: (4.9)By comparing the coefficients of t n on the right hand sides of the last two equations,we arrive at the desired result.

Theorem 3 . 1 .
The following implicit summation formulae for Lagrange-based Apostol type Hermite polynomials T

( 4 . 4 )
By comparing the coefficients of t n on the right hand sides of the last two equations, we arrive the desired result.

Remark 4 . 1 .
For α = 1, the above result reduces to n m=0 a n−m b m T
).The explicit representations of H E