On the confluent Euler-Poisson-Darboux equation and the Toda equation (Q1061938)

The papers under review deal with the representation of solutions of the Toda equation, i.e. of the partial differential-difference equation \[ (1)\quad (\partial^ 2/\partial x\partial y)\log t_ n=t_{n+1}t_{n- 1}/t^ 2_ n. \] There are elementary functions \[ (2)\quad t_ n=F_ n\exp ((\alpha -n)xy),\quad n=0,1,...,\quad F_{n+1}F_{n-1}=(\alpha - n)F^ 2_ n,\quad F_ 0=F_ 1=1, \] or \[ (3)\quad t_ n=F_ n(x- y)^{-f_ n},\quad n=0,1,...,\quad f_ n=(n-\alpha)(n-\beta), \] \[ \alpha,\beta =const.,\quad and\quad F_{n+1}F_{n-1}=-f_ nF^ 2_ n,\quad F_ 0=F_ 1=1, \] which formally satisfies (1). With the aid of Bäcklund transformations one gets new solutions from (2) or (3) in the form \(\tau_ n=u_ nt_ n,\) where \(u_ n\) are particular solutions of the confluent Euler-Poisson-Darboux-equation or the Euler-Poisson- Darboux-equation respectively. Rational and hypergeometric generated solutions of these equations are treated then. The theorems, too complicated to be reproduced here, are given without proofs.