In 1893 Giuseppe Lauricella

Giuseppe Lauricella

Giuseppe Lauricella was an Italian mathematician who is remembered today for his contributions to analysis and the theory of elasticity....

 defined and studied four hypergeometric series

Hypergeometric series

In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...

 F_A, F_B, F_C, F_D of three variables. They are :


for |x₁| + |x₂| + |x₃| < 1 and


for |x₁| < 1, |x₂| < 1, |x₃| < 1 and


for |x₁|^½ + |x₂|^½ + |x₃|^½ < 1 and


for |x₁| < 1, |x₂| < 1, |x₃| < 1. Here the Pochhammer symbol

Pochhammer symbol

In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 (q)_i indicates the i-th rising factorial power of q, i.e.

These functions can be extended to other values of the variables x₁, x₂, x₃ by means of analytic continuation

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

.

Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named F_E, F_F, ..., F_T and studied by Shanti Saran in 1954 . There are therefore a total of 14 Lauricella–Saran hypergeometric functions.

Generalization to n variables

These functions can be straightforwardly extended to n variables. One writes for example


where |x₁| + ... + |x_n| < 1. These generalized series too are sometimes referred to as Lauricella functions.

When n = 2, the Lauricella functions correspond to the Appell hypergeometric series

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...

 of two variables as follows:


When n = 1, all four functions reduce to the Gauss hypergeometric function,

Integral representation of F_D

In analogy with Appell's function F₁

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...

, Lauricella's F_D can be written as a one-dimensional Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

-type integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 for any number n of variables:


This representation can be easily verified by means of Taylor expansion

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 of the integrand, followed by termwise integration. The representation implies that the incomplete elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

Π is a special case of Lauricella's function F_D with three variables: