Abstract |
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, p(n)(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, p(n)(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 is an element of N-0. For a few values of such parameters, the three-term recurrence relation\n (TTRR) xp(n) = p(n+1) + beta(n)p(n) + gamma(n)p(n-1), n >= 0,\n presents some index for which the coefficient gamma(n) = 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 gamma(n) vanishes, i.e., {n : gamma(n) = 0} not equal phi.\n 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. (C) 2011 Elsevier Inc. All rights reserved. |