Appell Functions and Multiple Averages

Abstract

If f is an analytic function of one complex variable, two applications of an averaging process produce from f an analogous analytic function $\mathcal{F}(b,Z,\beta )$ 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 $\beta $-parameter with each column. Special cases of $\mathcal{F}$ include the Appell functions $F_1$, $F_2$, and $F_3$ and the Lauricella functions $F_B$ and $F_D$. Transformations of these functions are shown to be equivalent to the symmetry of $\mathcal{F}$ under permutations of the rows or columns of Z. Differential equations, series expansions, and a Cauchy integral formula are given for $\mathcal{F}$. A different type of multiple average, obtained by averaging a function of several complex variables $f(z_1 , \cdots ,z_n )$ 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 La...