- Lauricella hypergeometric series
In 1893
G. Lauricella defined and studied fourhypergeometric 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).
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