symmetry S S Article A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials Ghulam Muhiuddin 1, * , Waseem Ahmad Khan 2 , Ugur Duran 3 and Deena Al-Kadi 4   Citation: Muhiuddin, G.; Khan, W.A.; Duran, U.; Al-Kadi, D. A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials. Symmetry 2021, 13, 648. https:// doi.org/10.3390/sym13040648 Academic Editor: Dorian Popa Received: 20 March 2021 Accepted: 8 April 2021 Published: 11 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia 2 Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia; wkhan1@pmu.edu.sa 3 Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey;ugur.duran@iste.edu.tr 4 Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi Arabia; d.alkadi@tu.edu.sa * Correspondence: chistygm@gmail.com Abstract: 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. Keywords: hypergeometric Bernoulli polynomials; Lagrange polynomials; hypergeometric Lagrange– Hermite–Bernoulli polynomials; confluent hypergeometric function; special polynomials 1. Introduction Special polynomials (like Bernoulli, Euler, Hermite, Laguerre, etc.) have great impor- tance in applied mathematics, mathematical physics, quantum mechanics, engineering, and other fields of mathematics. Particularly the family of special polynomials is one of the most useful, widespread, and applicable families of special functions. Recently, the afore- mentioned polynomials and their diverse extensions have been studied and introduced in [1–14]. 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). (1) 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 1924, Nörlund [13] introduced the Bernoulli polynomials and numbers of order α :  t e t - 1  α e zt = e zt  e t -1 t  α = ∞ ∑ n=0 B (α) n (z) t n n! . (2) Symmetry 2021, 13, 648. https://doi.org/10.3390/sym13040648 https://www.mdpi.com/journal/symmetry