Lauricella Functions

Lauricella functions are generalizations of the Gauss hypergeometric functions to multiple variables. Four such generalizations were investigated by Lauricella (1893), and more fully by Appell and Kampé de Fériet (1926, p. 117). Let be the number of variables, then the Lauricella functions are defined by

			(1)
			(2)
			(3)
			(4)

A direct application of to products of Bessel functions of the first kind is given by the Weber-Schafheitlin integral.

The respective domains of absolute convergence of the defining series are

(5)

(Exton 1978, pp. 29-30).

For , the Lauricella function has the Euler-type integral representation

(6)

Along , all powers are principal. Initially, each is taken in , which keeps off the standard branch cut . Other values are obtained by analytic continuation, using consistent choices of the powers and, when necessary, deforming the integration contour to avoid singularities (Exton 1976, Ch. 5).

If , then these functions reduce to the Appell hypergeometric functions , , , and , respectively. If , all four become the Gauss hypergeometric function (Exton 1978, p. 29).

For generic parameters, the Horn systems associated with and have holonomic ranks , while the system associated with has rank (Banik and Bera 2026). Outside the domains above, the functions require analytic continuation. One numerical approach constructs the associated first-order Pfaffian system, restricts it to a one-dimensional path, and matches generalized power-series solutions using the Frobenius method. Banik and Bera (2026) describe a high-precision implementation of this approach for all four Lauricella families.