Appell Functions and Multiple Averages

If f is an analytic function of one complex variable, two applications of an averaging process produce from f an analogous analytic function F(b,Z,β ) depending on a rectangular matrix Z of complex variables and on two sets of complex parameters, one b-parameter being associated with each row of Z and one β-parameter with each column. Special cases of F include the Appell functions F₁, F₂, and F₃ and the Lauricella functions FB and FD. Transformations of these functions are shown to be equivalent to the symmetry of F under permutations of the rows or columns of Z. Differential equations, series expansions, and a Cauchy integral formula are given for F. A different type of multiple average, obtained by averaging a function of several complex variables f(z₁ , ⋯ ,zₙ ) with respect to each variable separately, is denoted by $F(B,Z)$, where B and Z are matrices of parameters and variables, respectively. The transformations of Lauricella’s FA are equivalent to permutation symmetries of $F(B,Z)$.