A characterization of some semi classical orthogonal polynomials. Issue 9 (1st September 2016)

• Record Type: Journal Article
• Authors: Griffin, James
• Is Part Of: Journal of difference equations and applications. Volume 22:Issue 9(2016)
• Page Start: 1261
• Page End: 1270
• Publication Date: 2016-09-01
• Subjects: Non linear difference equations -- orthogonal polynomials -- semi classical Hermite polynomials -- differential-difference equations; 33E30 -- 33D45; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2016.1191479
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : We show that if { P n } is a sequence of monic orthogonal polynomials satisfying a differential-difference equation of the form π ( x ) d d x P n ( x ) = b n P n ( x ) + ( c n x + d n ) P n - 1 ( x ) c n ≠ 0 then the orthogonal polynomial sequence is necessarily orthogonal to a well known semi classical generalization of the classical Hermite weight. If c n is allowed to be any real number then we show that the Laguerre polynomials also appear as a limiting case. The method is based on a derivation of a pair of non linear difference equations which are a special case of the asymmetric discrete Painleve IV equations. Such equations appear entirely as a consequence of the differential-difference equation together with the three term recurrence relation for a general sequence of orthogonal polynomials as opposed to previous derivations that stem from the consideration of a specific weight function.