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Grünbaum, F. Alberto; Vinet, Luc; Zhedanov, Alexei

Tridiagonalization and the Heun equation. (English) Zbl 1364.34126

J. Math. Phys. 58, No. 3, 031703, 12 p. (2017).
Summary: It is shown that the tridiagonalization of the hypergeometric operator \(L\) yields the generic Heun operator \(M\). The algebra generated by the operators \(L\;, M\; \text{and}\; Z = [L, M]\) is quadratic and a one-parameter generalization of the Racah algebra. The Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators \(L\) and \(M\). An interpretation in terms of the Racah problem for \(\operatorname{su}(1,1)\) and separation of variables in a superintegrable system are discussed.{
©2017 American Institute of Physics}

Cited in 25 Documents

MSC:

34M03 Linear ordinary differential equations and systems in the complex domain
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
47L30 Abstract operator algebras on Hilbert spaces
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Keywords:

Racah-Heun orthogonal polynomials
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References:

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