Dual Discrete Canonical Systems and Dual Orthogonal Polynomials

Abstract

The string equation

$$\frac{{d^2 \phi \left( {x,\lambda } \right)}}{{dx^2 }} = \lambda \rho ^2 \left( x \right)\phi \left( {x,\lambda } \right),\rho \left( x \right) > 0,0 \leqslant x \leqslant l$$

can be written in the form

$$\frac{{d^2 \phi \left( {x,\lambda } \right)}}{{dx^2 }} = \lambda \frac{{dM}}{{dx}}\phi \left( {x,\lambda } \right),$$

((0.1))

where

$$M\left( x \right) = \int\limits_0^x {\rho ^2 \left( t \right)dt.} $$

The equation

$$\frac{{d^2 \tilde \phi \left( {M,\lambda } \right)}}{{dM^2 }} = \lambda \frac{{dx}}{{dM}}\tilde \phi \left( {M,\lambda } \right)$$

((0.2))

is said to be dual to equation (0.1). The notion of a dual string was investigated by I.S. Kac and M.G. Krein [1]. Kac and Krein obtained the dual string equation from the original by interchanging the variables x and M(x). Let us add conditions

$$\phi \left( {0,\lambda } \right) = 1,\phi '\left( {0,\lambda } \right) = 0,$$

((0.3))

$$\tilde \phi \left( {0,\lambda } \right) = 0,\tilde \phi '\left( {0,\lambda } \right) = 1$$

((0.4))

to equations (0.1) and (0.2).

To Harry Dym, with whom I began working on the notion of duality, with attachment and friendship