The Associated Ultraspherical Polynomials and their q-Analogues

A sequence of polynomials { P n ( x )} is orthogonal if P n ( x ) is of precise degree n and there is a finite positive measure d μ such that 1.1 A necessary and sufficient condition for orthogonality [ 9 ] is that { P n ( x )} satisfies a three term recurrence 1.2 with 1.3 Given a sequence of orthogonal polynomials { P n ( x )} satisfying (1.2), the associated polynomials { P n ( γ ) ( x )}, γ > 0, are defined by 1.4 with P ( γ ) -1( x ) = 0, P 0 ( γ ) ( x ) = 1, when A n + γ and B n + γ are well-defined.