Filomat 32:20 (2018), 6903–6909 https://doi.org/10.2298/FIL1820903A Published by Faculty of Sciences and Mathematics, University of Niˇ s, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Miscellaneous Properties for a Class of Complex Functions Defined by Rodrigues Type Formula Rabia Aktas ¸ a , Fatma Tas ¸delen a a Ankara University, Faculty of Science, Department of Mathematics, 06100, Ankara, Turkey Abstract. In the present paper, we consider a class of holomorphic complex functions defined by Rodrigues type formula and we investigate some properties such as generating function and recurrence relations. Also, we discuss some applications of the given results. 1. Introduction Orthogonal polynomials in two complex variables were studied by many researchers. Zernike intro- duced 2D analogue of Legendre polynomials which are orthogonal on |z|≤ 1 and which arise on optical problems [20, 21]. The more general disc or Zernike polynomials were given in [13, 17] (see also [4]). Ito [12] defined the complex Hermite polynomials H m,n (z 1 , z 2 ) generated by (similar polynomials with other names and notations were defined by [16]) ∞  m,n=0 H m,n (z 1 , z 2 ) u m m! v n n! = e uz 1 +vz 2 -uv , which are studied several mathematical physicists, for example, they are applied to quantum optics and quasi probabilities and they are seen in combinatorics [14, 18, 19]. The Rodrigues formula is as follows H m,n (z 1 , z 2 ) = (-1) m+n e z 1 z 2 ∂ m+n ∂z n 1 ∂z m 2  e -z 1 z 2  . Also, q -analogues of the Ito’s polynomials were introduced by Ismail and Zhang in [11]. Moreover, there are many research papers for complex Hermite polynomials, we refer to [3, 6–9]. A brief review of orthogonal polynomials in two complex variables was given by Ismail and Zhang [10]. For other complex orthogonal polynomials and q-analogues of them, we refer to [10]. Other references on the 2D-orthogonal polynomials are [5], [15], [16], etc. In the previous works [1, 2], we have studied the family of real analytic functions defined by Rodrigues type formula and investigated their properties. Motivated by the class of real analytic functions, we consider 2010 Mathematics Subject Classification. Primary 33C45; Secondary 33C50 Keywords. Rodrigues formula, recurrence relation, generating function, 2D Hermite polynomials Received: 30 April 2018; Revised: 02 January 2019; Accepted: 03 January 2019 Communicated by Gradimir Milovanovi´ c and Miodrag Spalevi´ c Email addresses: raktas@science.ankara.edu.tr (Rabia Aktas ¸), tasdelen@science.ankara.edu.tr (Fatma Tas ¸delen)