Lauricella hypergeometric series

In 1893 G. Lauricella defined and studied four hypergeometric series of three variables. They are:

:$F\_A^\{(3)\}(a,b\_1,b\_2,b\_3,c\_1,c\_2,c\_3;x\_1,x\_2,x\_3)\; =\; sum\_\{i\_1,i\_2,i\_3=0\}^\{infty\}\; frac\{(a)\_\{i\_1+i\_2+i\_3\}\; (b\_1)\_\{i\_1\}\; (b\_2)\_\{i\_2\}\; (b\_3)\_\{i\_3\; \{(c\_1)\_\{i\_1\}\; (c\_2)\_\{i\_2\}\; (c\_3)\_\{i\_3\}i\_1!\; i\_2!\; i\_3!\}\; x\_1^\{i\_1\}x\_2^\{i\_2\}x\_3^\{i\_3\}$

:$F\_B^\{(3)\}(a\_1,a\_2,a\_3,b\_1,b\_2,b\_3,c;x\_1,x\_2,x\_3)\; =\; sum\_\{i\_1,i\_2,i\_3=0\}^\{infty\}\; frac\{(a\_1)\_\{i\_1\}\; (a\_2)\_\{i\_2\}\; (a\_3)\_\{i\_3\}\; (b\_1)\_\{i\_1\}\; (b\_2)\_\{i\_2\}\; (b\_3)\_\{i\_3\; \{(c)\_\{i\_1+i\_2+i\_3\}\; i\_1!\; i\_2!\; i\_3!\}\; x\_1^\{i\_1\}x\_2^\{i\_2\}x\_3^\{i\_3\}$

:$F\_C^\{(3)\}(a,b,c\_1,c\_2,c\_3;x\_1,x\_2,x\_3)\; =\; sum\_\{i\_1,i\_2,i\_3=0\}^\{infty\}\; frac\{(a)\_\{i\_1+i\_2+i\_3\}\; (b)\_\{i\_1+i\_2+i\_3\; \{(c\_1)\_\{i\_1\}\; (c\_2)\_\{i\_2\}\; (c\_3)\_\{i\_3\}i\_1!\; i\_2!\; i\_3!\}\; x\_1^\{i\_1\}x\_2^\{i\_2\}x\_3^\{i\_3\}$

:$F\_D^\{(3)\}(a,b\_1,b\_2,b\_3,c;x\_1,x\_2,x\_3)\; =\; sum\_\{i\_1,i\_2,i\_3=0\}^\{infty\}\; frac\{(a)\_\{i\_1+i\_2+i\_3\}\; (b\_1)\_\{i\_1\}\; (b\_2)\_\{i\_2\}\; (b\_3)\_\{i\_3\; \{(c)\_\{i\_1+i\_2+i\_3\}\; i\_1!\; i\_2!\; i\_3!\}\; x\_1^\{i\_1\}x\_2^\{i\_2\}x\_3^\{i\_3\}$

where the Pochhammer symbol $(a)\_\{i\}$ indicates the i-th rising factorial power of a, i.e. :$(a)\_\{i\}\; =\; a\; (a+1)\; dots\; (a+i-1).\; ,$ 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.

Generalization to n variables

These functions can be straightforwardly extended to $n$ variables. One writes for example

:$F\_A^\{(n)\}(a,b\_1,ldots,b\_n,c\_1,ldots,c\_n;x\_1,ldots,x\_n).$

When $n=2$ the Lauricella functions correspond to the Appell hypergeometric series of two variables as follows:

:$F\_Aequiv\; F\_2\; ,,\; F\_Bequiv\; F\_3\; ,,\; F\_Cequiv\; F\_4\; ,,\; F\_Dequiv\; F\_1.$

When $n=1$ all four functions reduce to the Gauss hypergeometric function:$,\_2F\_1(a;b;c;x).$

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).