Derivatives of any order of the confluent hypergeometric function F11(a,b,z) with respect to the parameter a or b

The derivatives to any order of the confluent hypergeometric (Kummer) function $F={}_{1}F_1(a,b,z)$ with respect to the parameter $a$ or $b$ are investigated and expressed in terms of generalizations of multivariable Kampé de Fériet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of $F$ with respect to $a$ are used to write the scattering wave function as a power series of the Sommerfeld parameter.