On singularities and solution of the extended hypergeometric differential equation

Abstract

It is well-known that a second-order differential equation with three regular singularities can be reduced to the Gaussian hypergeometric equation. A documented extension is the generalized hypergeometric equation, which also has three regular singularities, but is a differential equation of order higher than the second. In this paper a distinct generalization of the Gaussian hypergeometric equation is introduced (section 1), namely the extended hypergeometric equation, which is of second-order, and has two regular and one irregular singularity. It can be shown (Section 2) that it includes the Mathieu equation. The solution in power series, and with logrithmic singularities, are obtained in the neighborhood of the two regular singularities, as for the Gaussian type, with the diffrence (section 3) that the recurence formulas for the coefficients are not two-term but rather multiple-term. The solutions in the neighborhood of the irregular singularity at infinity are obtained (setion 4) by three methods, viz. normal integrals, Laurent series and transformation to Hill's equation. As a conclusion a physical problem is mentioned (section 5) in which the extended hypergeometric differential equation arises.