Orthogonal Polynomials, Asymptotics and Heun Equations (original) (raw)
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E. Heine in the 19th century studied a system of orthogonal polynomials and a related system was studied by C. J. Rees, in 1945. These are also known as elliptic orthogonal polynomials, since the moments of the weights maybe expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter k 2 , where 0 < k 2 < 1 is the τ function of a particular Painlevé VI, the special cases of which are related to enumerative problems arising from string theory. We show that the recurrence coefficients, denoted by β n (k 2 ), n = 1, 2, . . . ; and p 1 (n, k 2 ), the coefficients of x n−2 of the monic polynomials orthogonal with respect to the weight,
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Abstract. Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel–Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.