Appell series


In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by 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.

Definitions

The Appell series F1 is defined for |x| < 1, |y| < 1 by the double series:
where the Pochhammer symbol n represents the rising factorial:
where the second equality is true for all complex except.
For other values of x and y the function F1 can be defined by analytic continuation.
Similarly, the function F2 is defined for |x| + |y| < 1 by the series:
the function F3 for |x| < 1, |y| < 1 by the series:
and the function F4 for |x|½ + |y|½ < 1 by the series:

Recurrence relations

Like the Gauss hypergeometric series 2F1, the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's F1 is given by:
Any other relation valid for F1 can be derived from these four.
Similarly, all recurrence relations for Appell's F3 follow from this set of five:

Derivatives and differential equations

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 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 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.

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 F1:

Related series

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