J. Math. Computer Sci., 37 (2025), 226–235 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs The Appell sequences of fractional type Stiven Díaz Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla, Colombia. Abstract In the article, we explore a form of generalization of Appell polynomials stemming from fractional differential operators within the classical sense of Caputo and Riemann-Liuoville. To ascertain its generating function, we used the Mittag-Leffler function. Additionally, we propose a determinant form for this novel sequence family and derive general properties thereof. Keywords: The Appell polynomials, Caputo operator, Riemann-Liouville operator, Mittag-Leffler function. 2020 MSC: 11B83, 26A33, 11B68. ©2025 All rights reserved. 1. Introduction The Appell polynomials, denoted as {A n (x)} n∈N 0 , constitute a significant mathematical sequence in- troduced by the esteemed French mathematician Paul Appell (see [3]). These polynomials satisfy the differential equation: d dx A n (x)= nA n-1 (x), n ∈ N, (1.1) with A 0 (x) being a non-zero constant. Alternatively, the sequence can be elegantly expressed through the generating function: f(t)e xt = ∞ X n=0 A n (x) t n n! , where f is a formal power series in t. The Appell polynomials exhibit diverse properties that render them invaluable in the realm of mathe- matical analysis, particularly within the study of differential equations and related fields, as documented by Adel, Khan et al., and Nemati et al., [9, 11]. Prominent instances of polynomial sequences satisfying equation (1.1), or equivalently the recursive relations, encompass the well-known polynomials of Bernoulli and Euler. The exponential generating functions for the geometric polynomials of Bernoulli and Euler are expressed as follows (refer to [2]): te xt e t - 1 = ∞ X n=0 B n (x) t n n! , |t| < 2π, and 2e xt e t + 1 = ∞ X n=0 E n (x) t n n! , |t| <π. (1.2) Email address: sdiaz47@cuc.edu.co (Stiven Díaz) doi: 10.22436/jmcs.037.02.07 Received: 2024-05-04 Revised: 2024-07-22 Accepted: 2024-08-10