Generalized hypergeometric equations of non-Fuchsian type (Q1063762)

The differential equation \(z^ ny^{(n)}=\sum^{n}_{k=1}(a_ k+b_ kz^ q)z^{n-k}y^{(n-k)}\), Re(q)\(\geq 0\), which can be reduced to one of generalized hypergeometric type, has a regular singular point at the origin and an irregular one at infinity. This equation has been studied by the first author (unpublished lecture notes) in the special case where all but one of the constants, \(b_ i\), were zero. The present paper deals with the more general case where there are v nonzero values of the \(b_ i\), and, by assuming that none of the exponents at the origin are equal (mod 1), n formal power series solutions are obtained about the regular singular point, while (with similar restrictions on the exponents at infinity) n-v formal solutions of algebraic type and v of exponential type are obtained. The Stokes multipliers, by means of which the solutions at the origin are expressed as linear combinations of those at infinity, are determined explicitly. In concluding remarks, it is indicated that these ideas can be modified in the case where the restrictions on the exponents noted above are modified, resulting in logarithmic behavior of the solutions at the singular points.