1 GENERATING FUNCTIONS FOR SOME EXCEPTIONAL ORTHOGONAL POLYNOMIALS Corina Grosu Politehnica University of Bucharest, Bucharest, Romania E-mail: cgr90@yahoo.com Marta Grosu Politehnica University of Bucharest, Bucharest, Romania E-mail: marta_grosu@yahoo.com Abstract: Generating functions for orthogonal polynomials play a key role in statistical applications, especially in correlation testing. Our aim is to obtain such information for exceptional orthogonal polynomials which can afterwards be applied in statistical tests with missing data. Mathematics Subject Classification (2010): 33C45, 42C05, 60E10 Key words: exceptional orthogonal polynomials, generating functions, Adler – Crum transformation 1. Introduction Multi-indexed exceptional orthogonal polynomials S N n S n x P  )} ( { , are polynomials characterized by the following conditions [2], [3], [4], [5], [10], [11], [12]: i) the index set of the polynomial system N  S N , N  S N ii) the set S of the deleted states is a finite set of natural integers 1 }, ,.., { 1  i r m m m iii) the degree of each polynomial is 1 ,    l n N n S iii) they form a complete set of orthogonal polynomials with respect to a positive definite measure iv) they are eigenfunctions of a second order differential operator One of the main consequences of the previous hypotheses is the fact that the sequence S N n S n x P  )} ( { , does not contain polynomials of all the degrees. In fact, it either starts at some degree 1  l or has gaps containing the degrees 1 }, ,.., { 1  i r m m m , depending on the type of deletion: virtual state deletion or eigenstate deletion [10], [11], [12]. Another consequence resides in the difficulty in proving orthogonality and completeness of such a sequence [2], [3], [4]. Such results have been given in various settings for exceptional orthogonal polynomials belonging to the following families: i) through dual families for exceptional Charlier and Hermite polynomials, for exceptional Meixner and Laguerre polynomials and for exceptional Hahn and Jacobi polynomials [2], [3], [4] ii) directly for exceptional Hermite polynomials, exceptional Laguerre polynomials and exceptional Jacobi polynomials [10], [11], [12]. Remark. It can be proved that, by using properties of the respective polynomials, the form of the exceptional polynomials presented in the first category is equivalent with the form of the corresponding polynomials given in the second category (see also [10],[12]).