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Harmonic functions of \(\text{su}_q(2)\) for \(q\in \mathbb{R}\) and \(q\to S^1\). (English) Zbl 1042.81043

Summary: We show in this paper that a particular family of Askey-Wilson polynomials can be interpreted directly in the light of \(q\)-deformed \(su_q(2)\) algebras. This approach allows us to correct previous results concerning the \(q\)-Legendre functions investigated in Ya. L. Granovskij and A. S. Zhedanov [J. Phys. A. Math. Gen. 26, 4331–4338 (1993; Zbl 0854.33013)]. We also establish the orthonormalization and the special cases \(q\to 1\) (classical limit) and \(q\to\infty\) (asymptote). We conclude that these \(q\)-Legendre functions differ significantly from their classical counterparts only when \(q\) is in the vicinity of the unitary circle, where the singular points of the absolute value of these \(q\)-functions undergo a series of bifurcations.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics

Citations:

Zbl 0854.33013
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