On Co-recursive Orthogonal Polynomials

is equivalent to (1.1) with 6„ = 0 (w^2) and Pi(0)p^0. The condition b„ = 0 (w^2) suggests the symmetric case, (i.e.,P„( — x) = ( —l)"P„(x)) but this is denied by the condition Pi(0) ^0. (In fact, (1.2) shows that Pn( — r)^0 whenever Pn(r)=0.) It then seems natural to ask what relations exist between a set of polynomials satisfying (1.2) and the corresponding symmetric polynomials which would be obtained from the equivalent relation (1.1) if the condition Pi (0)^0 were replaced withPi(0)=0. We are thus led to consider the following more general situation. Given the Pn(x) satisfying (1.1) (without the restriction bn = 0), let the "co-recursive" polynomials P*(x)=P*(x, c) be defined by