On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series

The Kampé de Fériet double series $F_{1:1;1}^{1:1;1}$ is studied through the solution to the associated first-order nonhomogeneous differential equation. It is shown that the integral of $t^{β+l}M(\cdot;β;λt)M(\cdot;β;-λt)$ over $t\in[0,T]$, $T\geq0$, $l=0,1,\ldots$, $\Reβ+l>-1$, is a linear combination of functions $F_{1:1;1}^{1:1;1}$. The integral is a generalization of a class of so-called Coulomb integrals involving regular Coulomb wave functions.

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