where E n H ( x ; λ ) and G n H ( x ; λ ) are the Hermite-based generalized Apostol-Euler polynomials and the Hermite-based generalized Apostol-Genocchi polynomials respectively, defined by the following generating functions:.
For a real or complex parameter α, the generalized Euler polynomials of degree n are defined by the following generating functions: ∑ n = 0 ∞ E n ( x ) t n n !
where H k ( x + j y ; ξ y ; 1, b, b ; λ y ) = E n ( x + j y ; b ; λ y ), where E n ( x ; a, b, c ) denotes the generalized Euler polynomials, which are defined by means of the following generating function: ( t b t − a t ) c x t = ∑ n = 0 ∞ E n ( x ; a, b, c ) t n n !
where n ∈ Z + and B n, χ denotes the usual generalized Bernoulli numbers, which are defined by means of the following generating function (see [1 22]): ∑ a = 0 f − 1 χ ( a ) e a t t e f t − 1 = ∑ n = 0 ∞ B n, χ t n n !
Let a, b, c ∈ R +, a ≠ b, x ∈ R. The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function: ( a t − u λ b t − u ) α c x t = ∑ n = 0 ∞ H n ( x ; u ; a, b, c ; λ ) t n n !. (2).
Generalized Genocchi polynomials are defined by means of the following generating function: (2.22).
The following generating relation is straightforward: ∑ k = 0 ∞ M k ( n ) t k k !
Next we introduce the following generating function (A x)=xB x)).
with the following generating function: exp ( x t + y t 2 ) = ∑ n = 0 ∞ t n n !
which are defined by means of the following generating function: ( λ e t − 1 ) v v !
The Apostol-Bernoulli polynomials B n ( x ; λ ) ( λ ∈ C ) are defined by means of the following generating function: (1.6).
