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

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

REFERENCES:

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. Palermo 7, 111-158, 1893.

Srivastava, H. M. and Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Chichester, England: Ellis Horwood, 1985.