New properties of hypergeometric series derivable from Feynman integrals II. A generalisation of the H function

For pt. I, see ibid., vol.20, p.4109 (1987). Further examples of the use of Feynman integrals enable the derivation of new properties of hypergeometric series including new analytic continuation formulae for a generalised hypergeometric series and for a Kampe de Feriet function. This motivates the derivation of two new summation formulae for a generalised hypergeometric series and furthermore leads to a natural generalisation of the H function. While the latter, as is well known, contains as particular cases most of the special functions of applied mathematics, it does not contain some of importance, for instance the Riemann zeta function nor indeed any polylogarithm. Our generalisation of the H function does contain the polylogarithm; it also contains the exact partition function of the Gaussian model from statistical mechanics. Another new result is the simple summation formula 3 F 2 (1,1,3/2;2,2; x)=-(-) -1 ln((1+(1-x) 1 2/)/2).