The hypergeometric function, the confluent hypergeometric function and WKB solutions
Take the Gauss hypergeometric equation and instead of parameters \(a,b,c\) let us write \(a=\alpha_0+\alpha \eta, b=\beta_0+\beta\eta, c=\gamma_0+\gamma\eta\) and instead of the unknown function \(w\) let us take \(w=x^{-c/2}(1-x)^{(-1/2)(a+b-c+1)}\psi\). Then the equation is written in the form \[ (-\frac{d^2}{dx^2}+\eta^2Q)\psi=0,\quad Q=\sum_{j=0}^N \eta^{-j}Q_j(x). \] To such an equation the WKB analysis can be applied. It gives formal solutions that using the Borel summation can be transformed to analytic solutions. In the paper under review a relation between these solutions and the classical solutions of the Gauss equation is established. Also analogous results for the equation for the function \(F_{1,1}\) are obtained.
confluent hypergeometric differential equation
scientific article; zbMATH DE number 7442181
scientific article