Anatomy of a q-generalization of the Laguerre/Hermite Orthogonal Polynomials

Chan, Chuan-Tsung; Liu, Hsiao-Fan

doi:10.48550/arxiv.1612.09039

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“…The correspondence inspired by more general orthogonal polynomials is not known yet, but it probably could be constructed. Another direction is to construct a generalization of these relations to the q-deformed case even in the case of the q-Laguerre polynomials [103].…”

Section: Discussionmentioning

confidence: 99%

Orthogonal Polynomials in Mathematical Physics

Chan

Mironov

Морозов

et al. 2018

Ludwig Faddeev Memorial Volume

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This is a review of (q-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of (q-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra.The reduction to polynomials happens whenever any of the parameters a is negative integer, a = −n since then (a) k = Γ(a+k) Γ(a) vanishes for all k > n. The resultant from this truncation is a polynomial both in all other parameters a and in the argument z. The difference is only that the dependence on a's is "quantized", while on z is classical, in fact, z can be considered as an infinitely-large parameter so that asymptotically (z) k ∼ z k . Accordingly, a class of polynomials as functions of z is nowadays called "very classical", while those considered as functions of a, just "classical" (with "quantum" in this context referring to q-deformations).Thus, natural for physical applications are the hypergeometric functions, and natural for hypergeometric functions is a reduction to polynomials. However, what is much less natural for these polynomials is orthogonality. Orthogonal polynomials are distinguished by the peculiar property known as the 3-term relation, while natural for the hypergeometric polynomials is to satisfy a p-term relation, with p depending on the number of a and b parameters, i.e. on the type (r, s) of the hypergeometric function r F s . As we explain in the paper, for the most interesting case of r = s+ 1 (other values of r − s can be achieved by taking certain limits of these reference functions) the parameter p is equal to 3 for s ≤ 3, while in general it is rather equal to 2s − 3, i.e. exceeds 3 for large s. Moreover, if one insists on polynomiality of the z-dependence, i.e. on the "very classical" polynomials, the p = 3 bound actually shifts further down to s = 2, i.e. to the ordinary hypergeometric functions. It is this property that explains at the quantum level the somewhat mysterious limitations on s allowing appearance of orthogonal polynomials in the celebrated Askey scheme [5,6]. This, in turn, raises a very interesting question of the role of polynomials with p-term relations: clearly it should exist and be quite important in quantum field theory.A natural point to look at from this perspective would be matrix-model applications of orthogonal polynomials, where the natural conjecture could be that the orthogonality (p = 3) is related to integrability, the basic property of matrix model partition functions. Perhaps, increasing p could be still another natural deformation of integrability.Another side of the story at p = 3 concerns the orthogonality measure of the polynomials. Extremely interesting is the case when the weight function reduces to a finite sum of delt...

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Section: Discussionmentioning

confidence: 99%