Quest Journals Journal of Research in Applied Mathematics Volume 7 ~ Issue 1 (2021) pp: 01-04 ISSN(Online) : 2394-0743 ISSN (Print): 2394-0735 www.questjournals.org * Corresponding Author: Mortada S. Ali 1|Page Research Paper Orthogonality of Some Related Polynomials to Three-parameter Mittag–Leffler Function Hussam M. Gubara 1 , Alshaikh.A.Shokeralla 2 , Mortada S. Ali 3 1 Department of Mathematics, Faculty of Mathematical Sciences and Statistics, AlNeelain University, Khartoum, Sudan 2,3 Department of Mathematics, College of Science and Arts, Al-Baha University, Al-Makhwah, KSA Corresponding Author: Mortada S. Ali ABSTRACT: In this paper, the relationship between 3-parameter Mittag-Leffler function and Legendre polynomial is investigated. The determination of orthogonality of some special cases of 3-parameter Mittag- Leffler also is presented by using some properties Legendre polynomial and its generating function. KEYWORDS: Mittag–Leffler function , Legendre Polynomials , Orthogonal Functions. Received 03 Jan, 2021; Revised: 14 Jan, 2021; Accepted 16 Jan © The author(s) 2021. Published with open access at www.questjournals.org I. INTRODUCTION The Mittag–Leffler function was introduced by Gustaf Mittag-Leffler in 1903[1,2]. The fundamental Mittag-Leffler function is a generalization of the expansion of the exponential function [3], it is important function and it has many applications in physics, engineering, biological fields and in another sciences[4-16]. The basic Mittag-Leffler function is defined as              A 2-parameter is generalization of    and it is defined as              and 3-parameter is generalization of    and it is defined by                   All these functions are very important in fractional calculus and mathematical methods, they have relations with some special functions, some of these relations are discussed in this paper. This paper is structured as follows: Section 2 represents the Legendre polynomials and its basic properties. In section 3 we will show how to represent Legendre function using the 3-parameter Mittag-Leffler function at    ,  and    , we will use this relation to show orthogonality of some special cases 3- parameter Mittag-Leffler function. II. LEGENDRE POLYNOMIALS Legendre polynomial [17,18] is defined by                           where