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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 n be the number of variables, then the Lauricella functions are defined by

 F_A^((n))(a,b_1,...,b_n;c_1,...,c_n;x_1,...x_n) 
 =sum((a,m_1+...+m_n)(b_1,m_1)...(b_n,m_n)x_1^(m_1)...x_n^(m_n))/((c_1,m_1)...(c_n,m_n)m_1!...m_n!) 
F_B^((n))(a_1,...,a_n,b_1,...,b_n;c;x_1,...,x_n) 
 =sum((a_1,m_1)...(a_n,m_n)(b_1,m_1)...(b_n,m_n)x_1^(m_1)...x_n^(m_n))/((c,m_1+...,m_n)m_1!...m_n!) 
F_C^((n))(a,b;c_1,...,c_n;x_1,...,x_n) 
 =sum((a,m_1+...+m_n)(b,m_1+...+m_n)x_1^(m_1)...x_n^(m_n))/((c_1,m_1)...(c_n,m_n)m_1!...m_n!) 
F_D^((n))(a,b_1,...,b_n;c;x_1,...,x_n) 
 =sum((a,m_1+...+m_n)(b_1,m_1)...(b_n,m_n)x_1^(m_1)...x_n^(m_n))/((c,m_1+...+m_n)m_1!...m_n!).

If n=2, then these functions reduce to the Appell hypergeometric functions F_2, F_3, F_4, and F_1, respectively. If n=1, all four become the Gauss hypergeometric function _2F_1 (Exton 1978, p. 29).


See also

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

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