Product formulas for multivariate and matrix-variate confluent hypergeometric functions

Abstract

In this note, we revisit one of Erdélyi's product formulas for the univariate confluent hypergeometric function ${}_1{F}_1$ and extend it to the multivariate confluent hypergeometric functions ${\Phi }_2^{(k)}$, ${\Psi }_2^{(k)}$, as well as to a certain matrix-variate confluent hypergeometric function. A useful connection related to fractional calculus is also given.