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