Discrete models of the harmonic oscillator and a discrete analogue of Gauss' hypergeometric equation
A bispectral problem is an eigenvalue problem of the form \[ (Af)(x,y)=\lambda(y) f(x,y),\qquad (Bf)(x,y)=\mu(x) f(x,y), \] where \(A\) is a linear operator acting on the \(x\)-variable and \(B\) acts on the \(y\)-variable. In the original formulation of the problem [\textit{H. Duistermaat} and \textit{F. A. Grünbaum}, Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)] \(A\) is a Schrödinger operator and \(B\) an arbitrary linear ordinary differential operator. Recently, cases when one or both operators are difference operators have received much attention. In [\textit{F. A. Grünbaum} and \textit{L. Haine}, Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 26, 143-172 (1997; Zbl 0868.35116)] the case when \(A\) is a tridiagonal difference operator and \(B\) a second order differential operator was considered. It was shown that the solutions may be described in terms of solutions to Gauss's hypergeometric differential equation. In the present paper, both \(A\) and \(B\) are tridiagonal matrices and the spectrum of \(A\) is assumed to be linear. That is, the problem \[ a_{i+1} f_{i+1,j}+a_i f_{ij}+a_{i-1} f_{i-1,j}=j f_{ij}, \] \[ b_{j+1} f_{i,j+1}+b_j f_{ij}+b_{j-1}f _{i,j-1} =\mu_i f_{ij} \] with \(f\) a doubly infinite matrix is considered. It is shown that \(\mu_i\) must be a second order polynomial in \(i\), and that the remaining data may be described in terms of the solutions to a difference equation of the form \[ w_{j+1}+p_2(j)w_j+p_4(j)w_{j-1}=0, \] with \(p_k\) polynomials of degree \(k\) of a certain, rather complicated, form. The latter equation is viewed as a discrete analogue of Gauss's hypergeometric equation.
scientific article; zbMATH DE number 1757281