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Continuous q-Hermite polynomials when $$q>1$$. (English) Zbl 0694.33006
q-Series and partitions, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 18, 151-158 (1989).
[For the entire collection see Zbl 0687.00005.]
If $$x\to ix$$ in the recurrence relation for the continuous q-Hermite polynomials of Rogers, the resulting polynomials are orthogonal on the real line with respect to a positive measure when $$q>1$$. Some explicit orthogonality relations are given here. The moment problem is indeterminate so there are many positive measures which can be used. One of these orthogonalities led to a new q-beta integral with four free parameters.
Reviewer: R.Askey

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
##### Keywords:
q-Hermite polynomials; q-Hermite polynomials of Rogers