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On a general system of orthogonal q-polynomials Jesus S~Inchez-Dehesa (*) ABSTRACT A general set of orthogonal q-polynomials (Pro (x); m = 0, 1, 2, ..., N} is introduced and charac- terized by its three-term recursion relation. This set unifies many of the different known systems of orthogonal q-polynomials, e.g. the Stieltjes-Wigert polynomials and their several generaliza- tions, the Brenke-Chihara polynomials, the A1 Salam-Carlitz polynomials, the A1 Salam-ehihara polynomials, .... Compact expressions of the moments of the asymptotical density of zeros of this global set of q-polynomials are explicitly found in terms of the coefficients of the three-term recurrence relation. As an example the asymptotical density of zeros of the known, above-men- tioned systems of orthogonal q-polynomials are calculated through its moments. 1. INTRODUCTION Very early Heine [1], Stieltjes [2] and Szeg5 [3] realized the close connection between the orthogonal q-polynomials (henceforth OqP's, in short) and the theta-functions, elliptic functions and continued frac- tions. Later the OqP's have become important by them- selves as is shown by the large number of papers in the literature which study their properties. Many different systems of OqP's have been proposed, e.g. [4] te [I6]. The purpose of this paper is three-fold. First in an attempt to unify the rich variety of families of OqP's existing in the literature, a general set of orthogonal q-polynomials {Pm(x) )N= o is introduced. This set is deFmed by means of its three-term recursion relation. Particular examples of this set are : The Stieltjes-Wigert polynomials [2,15,16] and its several generalizations [10, 11], the Brenke-Chihara polynomials [7, 8], the Wall polynomials [14], the A1 Salam-Carlitz polyno- mials [4], the A1 Salam-Chihara polynomials [5] ..... Secondly average properties of zeros of the new sys- stem of OqP's are investigated. Both for the sake of convenience in many cases and in order to go as far as possible in the description of the distribution of zeros, we deal here not only with the distribution density of zeros PN(X) of the polynomials PN(X) and its asympto- tical (i.e. large N limit)p(x) but also with the density function ~N(X) -PN(x/qcN), c being any non-vanish- ing real constant, and its asymptotical limit ~(x). The utmost effort has been concentrated on calculating the moments of the function ~(x) explicitly in terms of the coefficients of the three-term recursion relation. The main results are given in theorems I, II, III and IV. Finally these four theorems are applied to give explicit expressions of the moments of the asymptotical density ~(x) for the OqP sets of the above.mentioned authors. To the best of information of the author this is the first time that any properties for the zeros of these q-polynomials are given. This paper is divided as follows : in section 2the general system (Pm(x) } of OqP's is characterized by giving its three-term recurrence relation, and several preliminaries are written. In particular it is briefly shown how to get the moments of the density of zeros of the polynomials Pm(x) by using a method recently developed by the author [17,18]. In section 3 the moments of the distribution densities PN(X), p(x) and especially of ~(x) are explicitly found for the q-polynomials Pm(x) except for a special type to which section 4 is dedicated. Sec- tion 5 contains applications of theorems I, II, III and IV obtained in the two previous sections to several known families of OqP's. Sections 6 and 7 contain some conclusions and references. 2. DEFINITION AND PRELIMINARIES We shall define the new system of orthogonal q-polyno- mials by means of the recurrence relation b 2 Pm (x)= (X-am) Pm-1 (x) - m-1 Pm-2 (x) (1) P_I (x) = o; P0(x)= 1; m= 1, 2, 3 .... with the coefficients a m and b m given as follows (*) J. S~inchez-Dehesa, Departamento de Ffsica Nuclear, Facultad de Ciencias, Universidad de Granada, Granada, Spain. Journal of Computational and Applied Mathematics, volume 5, no 1, 1979. 37