Method of continual addition theorems and integral relations between the Coulomb functions and the Appell function $F_1$

The paper considers a function $A$ introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell's hypergeometric function $F_1$. The resulting formula for the transformation of the function $A$ is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier–Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function $F_1$.