Some Extensions of Lardner’s Relations between ₀ F₃ and Bessel Functions

The even and odd parts of a hypergeometric ₚ Fq series are expressed as ₂ₚ F2q + 1-series, and inversely, a restricted ₂ₚ F2q + 1-series is expressed in terms of a pair of ₚ Fq-series. Generalizations are obtained by decomposing Meijer’s G-function into two G-functions which are not, however, always even or odd. More fundamentally, it is shown that any solution of a generalized hypergeometric differential equation with restricted parameters can be expressed in terms of solutions of an equation with order half as large. The results are illustrated by applications to Bessel functions, Kelvin functions, generalized Fresnel integrals, and restricted ₄ F₃-series with unit argument. A number of restricted G-functions are expressed in terms of more familiar functions. A reducibility criterion is used to identify cases in which fourth order differential equations governing vibrations of beams and deformations of shells can be solved in terms of Bessel functions or ₁ F₁-functions.