On certain results related to the hypergeometric function \(F_K\)
Gauss function has been generalized by many mathematicians in three ways: (i) by increasing the number of parameters as generalized hypergeometric function, (ii) by increasing the number of variables e.g. Appell's functions F1, F2, F3, F4, etc. (iii) by increasing both number of parameters and variables e.g. Kampe de Feriet function, [\textit{H. M. Srivastava} and \textit{P. W. Karlsson}, Multiple Gaussian hypergeometric series. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press: a division of John Wiley \& Sons. (1985; Zbl 0552.33001)]. These generalizations have been attraction of many applied scientist due to their applications. \textit{G. Lauricella} [Rend. Circ. Mat. Palermo 7, 111--158 (1893; JFM 25.0756.01)] has extended this function for three variables by defining fourteen functions and \textit{S. Saran} [Gaṇita 5, 77--91 (1954; Zbl 0058.29602)] investigated their analytic and asymptotic properties. One of the Saran's function FK has been used in the computation of a generalized Nordsieck integral [\textit{M. G. Bustamante} et al., Comput. Phys. Commun. 171, No. 1, 40--48 (2005; Zbl 1196.81053)] and has played an important role in \textit{T. P. Hutchinson}'s research work on compound gamma bivariate distribution [Metrika 28, 263--271 (1981; Zbl 0493.62016)]. It has also occurred in the investigation of propagator seagull diagram and shown that can be expressed explicitly as a sum of certain forms of the functions FK, F4, and 2F1. Saran's function Fk and its generalizations are used by some physicists for their study on conformal blocks: \textit{A. V. Belitsky} [Phys. Lett., B 780, 66--73 (2018; Zbl 1390.81564)] and \textit{V. Rosenhaus} [J. High Energy Phys. 2019, No. 2, Paper No. 142, 35 p. (2019; Zbl 1411.81188)]. Thus the importance of Saran's function Fk has motivated the authors to investigate further properties of this function. A new single contour integral representation has been established and the analytic continuation and asymptotics are obtained. A new integral identity has been proved using the fractional integration by parts and a number of interesting particular cases are discussed and established their relationship with \textit{L. Koschmieder}'s work [Acta Math. 79, 241--254 (1947; Zbl 0032.06501)].
scientific article; zbMATH DE number 7389230
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