Dirichlet averages of $x^t \log x$

A neglected class of special functions may be described as Dirichlet averages of $x^t \log x$ or equivalently as derivatives of hypergeometric R-functions with respect to the degree of homogeneity. Special cases include the derivative of a Legendre function with respect to the degree and the derivative of Gauss’s hypergeometric function with respect to a numerator parameter. There are connections with the logarithmic derivative of the gamma function, with Euler’s dilogarithm, and with ${}_3 F_2 ;$ some special cases of ${ }_3 F_2 ;$ are thereby evaluated. Applications include a two-point boundary-value problem, mean values, series expansions of elliptic integrals, integral tables, and several physics problems. The discussion of various properties emphasizes series expansions, quadratic transformations, inequalities, and evaluation of special cases, including certain cases of the derivative of a Legendre function with respect to the degree.