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# 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).

Appell Hypergeometric Function, Generalized Hypergeometric Function, Horn Function, Kampé de Fériet Function

This entry contributed by Ronald M. Aarts

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## 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.

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Lauricella Functions

## Cite this as:

Aarts, Ronald M. "Lauricella Functions." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LauricellaFunctions.html