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).
Appell, P. and Kampé de Fériet, J. Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite.
Paris: Gauthier-Villars, 1926.Erdélyi, A. "Hypergeometric
Functions of Two Variables." Acta Math.83, 131-164, 1950.Exton,
H. Ch. 5 in Multiple
Hypergeometric Functions and Applications. New York: Wiley, 1976.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.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
,
then these functions reduce to the Appell
hypergeometric functions 
, 
, 
, and 
, respectively. If 
, all four become the Gauss hypergeometric function 
(Exton 1978, p. 29).
