Orthogonal Polynomials Defined by a Recurrence Relation

R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation xp„-,(x) = -^ p„(x) + an_xPn_x(x) + -^pn-2(x) In tn-\ and (-0 then the logarithm of the absolutely continuous portion of the corresponding weight function is integrable. The purpose of this paper is to prove R. Askey's conjecture and solve related problems. Let a be a nondecreasing function defined on the real line. Such a function a is called weight function if it takes infinitely many values and all its moments are finite.2 For a given weight a there exists a unique system of polynomials {pn(da)}TM=0 such that p„(da, x) = yn(da)x" + . . . (y„ > 0) and /oo PnPmd0i = 8„m. m These orthogonal polynomials satisfy the recurrence formula yn_x(da) xp*-\ida' x) = --TTVPn(d», x) + a„_,(da)p„_,(da, x) In \""/ :t~¿daj p*-¿da>*} (1) for n = 1, 2, . . . where/?0 = Yo'/'-i = 0 and aÁda) = ( XPn(da' x) da(x)J — (VI Received by the editors January 16, 1978 and, in revised form, July 7, 1978. AMS (MOS) subject classifications (1970). Primary 42A52. 1 Research sponsored by the National Science Foundation under grant No. MCS75-06687. 2Many authors reserve the term "weight" for absolutely continuous a's. The present paper deviates from the norm in this regard. © 1979 American Mathematical Society 0002-9947/79/0000-0269/S05.00 369 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use