TY - JOUR
T1 - Orthogonality of q-polynomials for non-standard parameters
AU - Costas-Santos, R.S.
AU - Sánchez-Lara, J.F.
JO - Journal of Approximation Theory
VL - 163
IS - 9
SP - 1246
EP - 1268
PY - 2011
DA - 2011/09/01/
SN - 0021-9045
DO - https://doi.org/10.1016/j.jat.2011.04.005
UR - https://www.sciencedirect.com/science/article/pii/S0021904511000712
KW - -orthogonal polynomials
KW - Favard’s theorem
KW - -Hahn tableau
KW - -Askey tableau
AB - q-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), is known only when the product of any two parameters a,b,c,d is not a negative integer power of q. Also, the orthogonality of the big q-Jacobi, pn(x;a,b,c;q), is known when a,b,c,abc−1 is not a negative integer power of q. In this paper, we obtain orthogonality properties for the Askey–Wilson polynomials and the big q-Jacobi polynomials for the rest of the parameters and for all n∈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=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 γn vanishes, i.e., {n:γn=0}≠0̸. We also apply this result to the continuous dual q-Hahn, big q-Laguerre, q-Meixner, and little q-Jacobi polynomials, although it is also applicable to any family of orthogonal polynomials, in particular the classical orthogonal polynomials.
ER -