New properties of hypergeometric series derivable from Feynman integrals. I. Transformation and reduction formulae

A study in the statistical mechanics of phase transitions has involved certain seemingly simple Feynman integrals which have yielded surprisingly significant mathematical results. The integrals enable the derivation of new transformation, summation and reduction formulae for some single and multiple variable hypergeometric series. The main result reported in this paper is a transformation formula, a special case of which is a summation formula for a symmetric combination of two non-terminating Clausen's hypergeometric series of unit argument.