<i>q</i>-Konhauser polynomials

Abstract. Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also discuss several other results based on the properties of the polynomials: the equivalence between the StieltjesWigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szegö polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

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“…We rely for this on formula (4.2) from [26]. The expression for Z is essentially property (4.1) in [26].…”

Section: Mathematical Properties Of the Biorthogonal Polynomialsmentioning

confidence: 99%

“…Moreover γ nn = ζ nn = λ n , by matching the dominant coefficients on both sides. Note that equation (4.2) in [26] contains a typo as it does not fulfill this last condition (it would give ζ nn = 1). And of course one has ζ nj (1) = γ nj (1) = δ nj .…”

Section: Mathematical Properties Of the Biorthogonal Polynomialsmentioning

confidence: 99%

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Dolivet

Tierz

2007

Journal of Mathematical Physics

108

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“…which were considered by Al-Salam and Verma [1], who proved that (q −nk ; q k ) j (αq; q) kj (xq) kj (q k ; q k ) j (2.5) and …”

Section: Remark 1 If We Take the Weight Functionmentioning

confidence: 94%

“…Jain and Srivastava [4] obtained linear and bilinear generating functions for the polynomials Z (α) n (x, k; q), which were defined in Section 2 (see also [1,Section 4]). In this section, we obtain some further properties of the q-Konhauser polynomials Z (α) n (x, k; q).…”

Section: Further Properties Of the Q-konhauser Polynomialsmentioning

confidence: 99%

“…In 1983, Al-Salam and Verma [1] constructed some q-extensions of the polynomials Y α n (x; k) and Z α n (x; k), which they called the q-Konhauser polynomials (see also [2]). More recently, Jain and Srivastava [4] derived linear and bilinear generating functions for one of these q-Konhauser polynomials and also suggested an alternative pair of q-Konhauser biorthogonal polynomials (see, for details, [4, pp.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Some properties of q-biorthogonal polynomials

Şekeroğlu

Srivástava

Taşdelen

2007

Journal of Mathematical Analysis and Applications

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Almost four decades ago, Konhauser introduced and studied a pair of biorthogonal polynomials Y α n (x; k) and Z α n (x; k) α > −1; k ∈ N := {1, 2, 3, . . .} , which are suggested by the classical Laguerre polynomials. The so-called Konhauser biorthogonal polynomials Z α n (x; k) of the second kind were indeed considered earlier by Toscano without their biorthogonality property which was emphasized upon in Konhauser's investigation. Many properties and results for each of these biorthogonal polynomials (such as generating functions, Rodrigues formulas, recurrence relations, and so on) have since been obtained in several works by others. The main object of this paper is to present a systematic investigation of the general family of q-biorthogonal polynomials. Several interesting properties and results for the q-Konhauser polynomials are also derived.

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