A special family of non-symmetric semi-classical forms of class one

Abstract

AbstractA form (linear functional) v is called regular if there exists a sequence of polynomials $$\{S_{n}\}_{n\ge 0}$$ { S n } n ≥ 0 , $$\deg (S_{n})=n$$ deg ( S n ) = n , which is orthogonal with respect to v. $$\{S_{n}\}_{n\ge 0}$$ { S n } n ≥ 0 is fully characterized by the following recurrence relation: $$S_{n+2}(x)=(x-\beta _{n+1})S_{n+1}(x)-\gamma _{n+1}S_{n}(x)$$ S n + 2 ( x ) = ( x - β n + 1 ) S n + 1 ( x ) - γ n + 1 S n ( x ) , $$n\ge 0$$ n ≥ 0 , with $$S_{0}(x)=1$$ S 0 ( x ) = 1 , $$S_{1}(x)=x-\beta _{0}$$ S 1 ( x ) = x - β 0 and $$\gamma _{n+1}\ne 0$$ γ n + 1 ≠ 0 , $$n\ge 0$$ n ≥ 0 . Such a form v is said be semi-classical if there exist polynomials $$\Psi (x)$$ Ψ ( x ) and $$\Phi (x)$$ Φ ( x ) with $$\deg (\Psi )\ge 1$$ deg ( Ψ ) ≥ 1 such that $$(\Phi v)' +\Psi v=0$$ ( Φ v ) ′ + Ψ v = 0 . When v is semi-classical and regular, its corresponding polynomial sequences $$\{S_{n}\}_{n\ge 0}$$ { S n } n ≥ 0 are called semi-classical. In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients $$\beta _{n}$$ β n , $$\gamma _{n+1}$$ γ n + 1 , $$n\ge 0$$ n ≥ 0 of the semi-classical orthogonal polynomials sequences of class one when $$\beta _{n}=t_{n-1}-t_{n}$$ β n = t n - 1 - t n and $$\gamma _{n+1}=-t_{n}(c+t_{n})$$ γ n + 1 = - t n ( c + t n ) with $$t_{n}(c+t_{n})\ne 0$$ t n ( c + t n ) ≠ 0 $$n\ge 0$$ n ≥ 0 , $$t_{-1}=0$$ t - 1 = 0 and $$c\in \mathbb {C}-\{0\} $$ c ∈ C - { 0 } . There are essentially five canonical cases.