Orthogonality of qq-polynomials for non-standard parameters

qq-polynomials can be defined for all the possible parameters, but their orthogonality properties are unknown for several configurations of the parameters. Indeed, orthogonality for the Askey–Wilson polynomials, pn(x;a,b,c,d;q)pn(x;a,b,c,d;q), is known only when the product of any two parameters a,b,c,da,b,c,d is not a negative integer power of qq. Also, the orthogonality of the big qq-Jacobi, pn(x;a,b,c;q)pn(x;a,b,c;q), is known when a,b,c,abc−1a,b,c,abc−1 is not a negative integer power of qq. In this paper, we obtain orthogonality properties for the Askey–Wilson polynomials and the big qq-Jacobi polynomials for the rest of the parameters and for all n∈N0n∈N0. For a few values of such parameters, the three-term recurrence relation (TTRR) xpn=pn+1+βnpn+γnpn−1,n≥0, presents some index for which the coefficient γn=0γn=0, and hence Favard’s theorem cannot be applied. For this purpose, we state a degenerate version of Favard’s theorem, which is valid for all sequences of polynomials satisfying a TTRR even when some coefficient γnγn vanishes, i.e., {n:γn=0}≠0̸{n:γn=0}≠0̸.We also apply this result to the continuous dual qq-Hahn, big qq-Laguerre, qq-Meixner, and little qq-Jacobi polynomials, although it is also applicable to any family of orthogonal polynomials, in particular the classical orthogonal polynomials.