On generalized Heun equation with some mathematical properties

Nasser Saad

doi:10.14311/AP.2022.62.0165

Authors

Nasser Saad University of Prince Edward Island, Department of Mathematics and Statistics, 550 University Avenue, Charlottetown, PEI, Canada C1A 4P3.

DOI:

https://doi.org/10.14311/AP.2022.62.0165

Keywords:

Heun equation, confluent forms of Heun’s equation, polynomial solutions, sequences of orthogonal polynomials

Abstract

We study the analytic solutions of the generalized Heun equation, (α₀ + α₁r + α₂r² + α₃r³) y′′ + (β₀ + β₁r + β₂r²) y′ + (ε₀ + ε₁r) y = 0, where |α₃| + |β₂|≠ 0, and {α_i}³_i=0, {β_i}²_i=0, {ε_i}¹_i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple procedure based on a recurrence relation is introduced to evaluate these polynomial solutions explicitly. For α₀ = 0, α₁≠ 0, we prove that the polynomial solutions of the corresponding differential equation are sources of finite sequences of orthogonal polynomials. Several mathematical properties, such as the recurrence relation, Christoffel-Darboux formulas and the norms of these polynomials, are discussed. We shall also show that they exhibit a factorization property that permits the construction of other infinite sequences of orthogonal polynomials.

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