Analytical and numerical analysis of the generalized Shkarofsky function (Q1194588)

The functions under consideration may be defined as follows: \[ \int_ 0^ \infty (1+s)^{-q} (1+bs)^{-p} \exp\left[-\zeta s-{as \over{1+s}}\right]ds, \] where \(a\), \(b\), \(p\), \(q\), \(\zeta\) are complex variables. The authors derive, by standard methods, a number of properties of these functions, such as analytic continuation formulae, recurrence relations, and asymptotic expressions. The main objective is to obtain results that are useful for those values of the variables that appear in the particular application (propagation in a plasma) that gave rise to the study of these functions. (Accordingly, transformation to Srivastava's \(F^{(3)}\) is not relevant.) Finally, extensive comparisons between numerical and asymptotic results are carried out by the aid of quite a few graphs.