On two identities connected with generalized solutions to hyperbolic equations (Q2742999)

Two identities connected to obtain the generalized solutions to the hyperbolic equation \(u_{xx}+yu_{yy}+\alpha u_y=0\), where \(\alpha=-n+\alpha_0\), \(\alpha_0\in(0;1/2)\cup(1/2;1)\), and \(n=0,1,\dots\) are considered. For example, it is proved that \(F(1+\beta,2\beta,2+2\beta;z)+\dfrac{\beta}{2(1+2\beta)}zF(2+\beta,1+2\beta,3+2\beta;z)-\dfrac{\beta}{2(1+2\beta)}zF(1+\beta,1+2\beta,3+2\beta;z)=(1-z)^{-\beta}\), where \(\beta=\alpha-1/2\) and \(F\) is a hypergeometric function (see [\textit{H. Bateman} and \textit{A. Erdélyi}, Higher transcendental functions. Vol. I. (1953, rev. ed. 1981; Zbl 0542.33001)]).