A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

In this paper, the usual notations refer to the set of all complex numbers C, the set of real numbers R, the set of all integers Z, the set of all natural numbers N, and the set of all non-negative integers N 0 , respectively. The classical Bernoulli polynomials B n (x) are defined by t e t − 1 e xt = ∞ ∑ n=0 B n (x) t n n! (|t| < 2π).
Upon setting x = 0 in (1), the Bernoulli polynomials reduce to the Bernoulli numbers, namely, B n (0) := B n . The Bernoulli numbers and polynomials have a long history, which arise from Bernoulli calculations of power sums in 1713 (see [9]), that is m ∑ j=1 j n = B n+1 (m + 1) − B n+1 n + 1 The Bernoulli polynomials have many applications in modern number theory, such as modular forms and Iwasawa theory [11].
In this paper, we introduce the multivariable unified Lagrange-Hermite-based hypergeometric Bernoulli polynomials and investigate some of their properties. Then, we derive multifarious connected formulas involving the Miller-Lee polynomials, the Laguerre polynomials polynomials, the Lagrange Hermite-Miller-Lee polynomials.

Theorem 2.
The following summation formula: holds for n ∈ N 0 .
Proof. By using (13), we have which gives the asserted result (16).
We give the following theorem: The following summation formula: holds for n ∈ N 0 .
Proof. For α = 1 and x = 0 in (13), we have Comparing the coefficients of t n in both sides, we get the result (21).
We give the following derivative property: M,N,n;l 1 ,··· ,l r (x|x 1 , · · · , x r )t n = Proof. The proof is similar to Theorem 3.

Some Connected Formulas
The generation functions (13) and (14) can be exploited in a number of ways and provide a useful tool to frame known and new generating functions in the following way: As a first example, we set α = α 2 = 0, α 1 = m + 1, x 1 = 1 in (13) to get where G (m) n (x) are called the Miller-Lee polynomials (see [4]). Another example is the definition of higher-order hypergeometric Bernoulli-Hermite-
which by using binomial expansion takes the form which implies the asserted result (28).
Proof. On replacing x with x + y and α 1 with α 1 + m + 1, respectively, in (13), we have which yields the claimed result (29). n (x) holds: Proof. For α 1 = m + 1 and x 1 = 1 in (13), we have Multiplying both the sides by (1 − x 1 t) −α 1 , we have Comparing the coefficient of t n , we get the result (30). Now, we shall focus on the connection between the higher-order generalized hyper- where H B  n (x) (see [14]).

Conclusions
In this paper, we define the multivariable unified Lagrange-Hermite-based hypergeometric Bernoulli polynomials and investigate some of their properties. Then, we derive multifarious connected formulas involving the Miller-Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite-Miller-Lee polynomials. It is demonstrated that the proposed the method allows the derivation of sum rules involving products of generalized polynomials and addition theorems. We developed a point of view based on generating relations, exploited in the past, to study some aspects of the theory of special functions. The possibility of extending the results to include generating functions involving products of Lagrange-Hermite-based hypergeometric Bernoulli polynomials and other polynomials is finally analyzed.