Appell series
In mathematics, Appell series are a set of four hypergeometric series "F", "F", "F", "F" of two variables that were introduced by and that generalize Gauss's hypergeometric series "F" of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. Definitions. The Appell series "F" is defined for |"x"| < 1, |"y"| < 1 by the double series where formula_2 is the Pochhammer symbol. For other values of "x" and "y" the function  F can be defined by analytic continuation. It can be shown that Similarly, the function "F" is defined for |"x"| + |"y"| < 1 by the series and it can be shown that Also the function "F" for |"x"| < 1, |"y"| < 1 can be defined by the series and the function "F" for |"x"| + |"y"| < 1 by the series Recurrence relations. Like the Gauss hypergeometric series "F", the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's "F" is given by: Any other relation valid  for "F" can be derived from these four. Similarly, all recurrence relations for Appell's "F" follow from this set of five: Derivatives and differential equations. For Appell's "F", the following derivatives result from the definition by a double series: From its definition, Appell's "F" is further found to satisfy the following system of second-order differential equations: A system partial differential equations for "F" is The system have solution Similarly, for "F" the following derivatives result from the definition: And for "F" the following system of differential equations is obtained: A system partial differential equations for "F  is The system have solution Integral representations. The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only . However, discovered that Appell's F" can also be written as a one-dimensional Euler-type integral: This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration. Special cases. Picard's integral representation implies that the incomplete elliptic integrals "F" and "E" as well as the complete elliptic integral Π are special cases of Appell's "F":