On the definite integral of two confluent hypergeometric functions related to the Kamp\'e de F\'eriet double series

The Kamp\'{e} de F\'{e}riet 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 $t^{\beta+l}M(\cdot;\beta;\lambda t)M(\cdot;\beta;-\lambda t)$ over $t\in[0,T]$, $T\geq0$, $l=0,1,\ldots$, $\Re\beta+l>-1$, 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.