Orthogonal polynomial interpretation of Δ-Toda equations

Orthogonal Polynomial Eigenfunctions of Second-order Partial Differerential Equations

In this paper, we show that for several second-order partial differential equations L[u] = A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy +D(x, y)ux + E(x, y)uy = λnu which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas ...

Recurrence equations and their classical orthogonal polynomial solutions

The classical orthogonal polynomials are given as the polynomial solutions pnðxÞ of the differential equation rðxÞy00ðxÞ þ sðxÞy0ðxÞ þ knyðxÞ 1⁄4 0; where rðxÞ is a polynomial of at most second degree and sðxÞ is a polynomial of first degree. In this paper a general method to express the coefficients An; Bn and Cn of the recurrence equation pnþ1ðxÞ 1⁄4 ðAnxþ BnÞpnðxÞ Cnpn 1ðxÞ in terms of the g...

Equations of motion for zeros of orthogonal polynomials related to the Toda lattices

Orthogonal polynomials and the finite Toda lattice

The choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed. It is then shown that the same induced time dependence for finitely supported distributions which gives the ith KP flow under the dual isomorphism induces the ith flo...

Toda Systems and Hypergeometric Equations

This paper establishes certain existence and classification results for solutions to SU(n) Toda systems with three singular sources at 0, 1, and∞. First, we determine the necessary conditions for such an SU(n) Toda system to be related to an nth order hypergeometric equation. Then, we construct solutions for SU(n) Toda systems that satisfy the necessary conditions and also the interlacing condi...