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
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.
Appell, P. and Kampé de Fériet, J. Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite.
Paris, France: Gauthier-Villars, 1926.Banik, S. and Bera, S. "HyperPrecision:
A Mathematica Package for High-Precision Numerical Evaluation of Multivariate Hypergeometric
Functions." May 28, 2026. https://arxiv.org/abs/2605.30216.Erdélyi,
A. "Hypergeometric Functions of Two Variables." Acta Math.83,
131-164, 1950.Exton, H. "The Lauricella Functions and Their Confluent
Forms," "Convergence," and "Systems of Partial Differential Equations."
§1.4.1-1.4.3 in Handbook
of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs.
Chichester, England: Ellis Horwood, pp. 29-31, 1978.Exton, H. Ch. 5
in Multiple
Hypergeometric Functions and Applications. New York: Wiley, 1976.Lauricella,
G. "Sulla funzioni ipergeometriche a più variabili." Rend. Circ.
Math. Palermo7, 111-158, 1893.Srivastava, H. M. and
Karlsson, P. W. Multiple
Gaussian Hypergeometric Series. Chichester, England: Ellis Horwood, 1985.
be the number of variables, then the Lauricella functions are defined by
to products of Bessel
functions of the first kind is given by the Weber-Schafheitlin
integral.
,
then these functions reduce to the Appell
hypergeometric functions 
, 
, 
, and 
, respectively. If 
, all four become the Gauss hypergeometric function 
(Exton 1978, p. 29).
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.
