Lauricella hypergeometric series
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In 1893 G. Lauricella defined and studied four hypergeometric series of three variables. They are:
where the Pochhammer symbol (a)i indicates the i-th rising factorial power of a, i.e.
Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named and studied by Saran in 1954. There are therefore a total of 14 Lauricella-Saran hypergeometric functions.
[edit] Generalization to n variables
These functions can be straightforwardly extended to n variables. One writes for example
When n = 2 the Lauricella functions correspond to the Appell hypergeometric series of two variables as follows:
When n = 1 all four functions reduce to the Gauss hypergeometric function
[edit] References
- G. Lauricella: Sulle funzioni ipergeometriche a più variabili, Rend. Circ. Mat. Palermo, 7, p111-158 (1893).
- S. Saran: Hypergeometric Functions of Three Variables, Ganita, 5, No.1, p77-91 (1954).
