In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series 2F1 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.
For other values of x and y the function F1 can be defined by analytic continuation.
Similarly, all recurrence relations for Appell's F3 follow from this set of five: For Appell's F1, the following derivatives result from the definition by a double series: From its definition, Appell's F1 is further found to satisfy the following system of second-order differential equations: A system partial differential equations for F2 is The system have solution Similarly, for F3 the following derivatives result from the definition: And for F3 the following system of differential equations is obtained: A system partial differential equations for F4 is The system has solution The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn et al. 2015, §9.184).
However, Émile Picard (1881) discovered that Appell's F1 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.