Lauricella hypergeometric series

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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.

[edit] 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_A\equiv F_2 ,\, F_B\equiv F_3 ,\, F_C\equiv F_4 ,\, F_D\equiv F_1.

When n = 1 all four functions reduce to the Gauss hypergeometric function

\,_2F_1(a;b;c;x).

[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).