Linear differential relations between solutions of a class of Euler-Poisson-Darboux equations (Q2567827)

An Euler-Poisson-Darboux equation is represented in one of the forms: \[ \frac{\partial^2u}{\partial r^2} +\frac{\alpha}{r}\frac{\partial u}{\partial r} +\frac{\partial^2u}{\partial z^2}=0,\; \frac{\partial^2u}{\partial r^2} +\frac{\alpha}{r}\frac{\partial u}{\partial r} -\frac{\partial^2u}{\partial z^2}=0,\; \frac{\partial^2u}{\partial\xi\partial\eta} +\frac{\alpha}{2(\xi+\eta)}\frac{\partial u}{\partial\xi} +\frac{\partial u}{\partial\eta}=0, \] where \(\alpha\) is a real parameter. These equations specify three classes of Euler-Poisson-Darboux equations. Equations of one class are determined by the parameter \(\alpha\). In this paper, all linear first-order differential relations of the form \[ u^{(\beta)}=A(r,z)\frac{\partial u^{(\alpha)}}{\partial r} +B(r,z)\frac{\partial u^{(\alpha)}}{\partial z} +C(r,z)u^{(\alpha)} \] between solutions \(u^{(\alpha)}\) and \(u^{(\beta)}\) of Euler-Poisson-Darboux equations of one class are obtained. These relations are used to derive identities between Euler-Poisson-Darboux operators, recursion relations for Bessel functions, and general solutions of one-dimensional gas dynamics of a polytropic gas. This paper is an extended version of the paper [\textit{A. V. Aksenov}, Dokl. Math. 64, No. 3, 421--424 (2001; Zbl 1048.35049); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 381, No. 2, 176--179 (2001; Zbl 1028.35116)].