On the solvability of confluent Heun equation and associated orthogonal polynomials

The present paper analyze the constraints on the confluent Heun type-equation, $(a_{3,1}{r}^2+a_{3,2}r)y\mathrm{"}+(a_{2,0}{r}^2+a_{2,1}r+a_{2,2})y^{\mathrm{\prime }}-({\tau }_{1,0}r+{\tau }_{1,1})y=0,$ where $\mathrm{\mid }{a}_{3,1}{\mathrm{\mid }}^2+\mathrm{\mid }{a}_{3,2}{\mathrm{\mid }}^2\mathrm{\ne }0,$ and $a_{i,j},i=3,2,1,j=0,1,2$ 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 $V(r)=-V_0{\sinh }^p(r)\mathrm{/}{\cosh }^q(r),V_0>0,p\ge 0,q>p$. The associated orthogonal polynomials generated by the solutions of the Schr"odinger equation with the $(4,6)$-hyperbolic potential are constructed.