Linearization and connection coefficients of orthogonal polynomials. (English) Zbl 0766.33008
Let \(\{P_ n\}_{n=0}^ \infty\) be a system of orthogonal polynomials. R. Lasser observed that if the linearization coefficients of \(\{P_ n\}_{n=0}^ \infty\) are nonnegative then each of the \(P_ n\) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomials \(P_ n\) can be expressed in terms of \(Q_ n\) with nonnegative coefficients, where \(\{Q_ n\}_{n=0}^ \infty\) is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.
Reviewer: R.Szwarc (Wrocław)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
recurrence formulaReferences:
| [1] | Askey, R.: Orthogonal expansions with positive coefficients II. SIAM J. Math. Anal.2, 340-346 (1971). · doi:10.1137/0502031 |
| [2] | Askey, R.: Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM. 1975. · Zbl 0298.33008 |
| [3] | Chihara, T. S.: An Introduction to Orthogonal Polynomials. New York: Gordon and Breach. 1978. · Zbl 0389.33008 |
| [4] | Karlin, S., McGregor, J.: The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc.85, 489-546 (1957). · Zbl 0091.13801 · doi:10.1090/S0002-9947-1957-0091566-1 |
| [5] | Lasser, R.: Orthogonal polynomials and hypergroups: the symmetric case. Preprint. · Zbl 0804.42013 |
| [6] | Micchelli, C. A.: A characterization of M. W. Wilson’s criterion for nonnegative expansions of orthogonal polynomials. Proc. Amer. Math. Soc.71, 69-72 (1978). · Zbl 0391.42018 |
| [7] | Nevai, P.: Orthogonal Polynomials. Mem. Amer. Math. Soc213 (1979). · Zbl 0405.33009 |
| [8] | Szwarc, R.: Connection coefficients of orthogonal polynomials. Canad. Math. Bull. To appear. · Zbl 0762.39010 |
| [9] | Szwarc, R.: Orthogonal polynomials and a discrete boundary value problem I. SIAM J. Math. Anal.23 (1992). To appear. · Zbl 0772.42013 |
| [10] | Szwarc, R.: Orthogonal polynomials and a discrete boundary value problem II. SIAM J. Math. Anal.23 (1992). To appear. · Zbl 0772.42014 |
| [11] | Wilson, M. W.: Nonnegative expansions of polynomials. Proc. Amer. Math. Soc.24, 100-102 (1970). · Zbl 0184.09603 · doi:10.1090/S0002-9939-1970-0287244-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
