Orthogonal polynomials and hypergroups. II. The symmetric case

Abstract

Abstract. The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on No . If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monk recursion formula and random walk polynomials. Polynomial hypergroupsIn [18] we demonstrated a close relationship between certain hypergroups on No and certain orthogonal polynomial sequences. In this paper we discuss other basic properties concerning these hypergroups-we call them polynomial hypergroups-and give many examples based on certain classes of orthogonal polynomial sequences. Our main reference is [18]. Notation and results from [18] are used throughout.The best known class corresponding to a polynomial hypergroup are the Jacobi polynomials (PJT 'ß) (x))f?=Q fora>jff>-l,a + jff-r-l>0 (see [18, 3(a)]). The basic property (P) holds as is shown by Gasper [11]. In [12]Gasper also proved that (Pna \x))f¡Lo defines a dual hypergroup structure on the interval [-1, 1]. Recently, Connett and Schwartz [9] proved that the Jacobi polynomials with a > ß > -1 and either ß > -1/2 or a + ^ > 0 are the only ones which define a dual hypergroup structure on [- (see e.g. [20-23, 32-34]). We reveal further polynomial hypergroups and determine their recurrence relations, their Haar weights and specific properties of the corresponding polynomials. In the following we consider the case of even orthogonalization measures only. Also, we choose x = 1 as the point

Reiter?s Condition P1 and Approximate Identities for Polynomial Hypergroups

“…Polynomial hypergroups are a very interesting class since one can find hypergroups in this class which are quite different from groups, see for example [18]. In [9] we already studied polynomial hypergroups in view of the P 1 ð; MÞ condition.…”

“…To have a good reference and for the sake of completeness we recall the basic facts for polynomial hypergroups. For more details and the proofs we refer to [17] and [18].…”

“…There are many orthogonal polynomial systems which have this property (cf. [1], [11], [17], [18], [25], [26]). An easy calculation shows that for every orthogonal polynomial system we have hðnÞ ¼ gðn; n; 0Þ À1 .…”

On Some Classes of Polynomials Orthogonal on Arcs of the Unit Circle Connected with Symmetric Orthogonal Polynomials on an Interval

“…Finally, we would like to note that in [23] some interesting applications of chain sequences to SOPI were considered. Due to DG transformation from SOPI to OPC one should expect that the results of [23] could be applied to the polynomials orthogonal on the unit circle.…”

A new characterization of ultraspherical polynomials

“…The notion of convolution is motivated by the theory of polynomial hypergroups; compare [4]. Of course, here we do not suppose that the orthogonal polynomials R n (x) induce a polynomial hypergroup.…”