2015/08/31 by Saad, Nasser
#33C05 #33C15 #33C47 #34A05 #34B30 #34B60 #47E05 #81Q05 #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1509.00090
The present paper analyze the constraints on the confluent Heun type-equation, (a3,1r2+a3,2r)y"+(a2,0r2+a2,1r+a2,2)y'-(τ1,0r+τ1,1)y=0, where |a3,1|2+|a3,2|2≠ 0, and ai,j,i=3,2,1, j=0,1,2 are real parameters, to admit polynomial solutions. The necessary and sufficient conditions for the existence of these polynomials are given. A three-term recurrence relation is provided to generate the polynomial solutions explicitly. We, then, prove that these polynomial solutions are a source of finite sequences of orthogonal polynomials. Several properties, such as the recurrence relation, Christoffel-Darboux formulas and the moments of the weight function, are discussed. We also show a factorization property of these orthogonal polynomials that allow for the construction of other sequences of orthogonal polynomials. For illustration, we examines the quasi- exactly solvability of the (p,q)-hyperbolic potential V(r)=-V0\sinhp(r)/\coshq(r), V0>0, p≥ 0, q>p. The associated orthogonal polynomials generated by the solutions of the Schrödinger equation with the (4,6)-hyperbolic potential are constructed.