Appell hypergeometric partial differential equations

Mathematicians often speak of the unity of their subject. Such a unity from the 19th century origins in the history of linear ordinary differential equations, especially those closely connected with elliptic and modular functions. It is concerned with the impact of group theoretical and geometrical ideas upon the problem of understanding the nature of the solutions to a differential equation (see [Kl2], [Kl3] and [G] as excellent surveys). The theory of hypergeometric functions has a long history (see [Ho1]). The first occurrence of hypergeometric series goes back to the 17th century. Euler was the first to consider the series as a function of one variable x: