Lauricella hypergeometric series explained

In 1893 Giuseppe Lauricella defined and studied four hypergeometric series F_A, F_B, F_C, F_D of three variables. They are :

F ^{( 3 )} _{A} (a,b _{1,b} _{2,b} _{3,c} _{1,c} _{2,c} _{3;x} _{1,x} _{2,x} _{3)} = \sum ^{inf ty} _{i _{1,i} _{2,i} _{3=0}} ((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 ^{i _{1}} _{1} x ^{i _{2}} _{2} x ^{i _{3}} _{3}

for |x₁| + |x₂| + |x₃| < 1 and

F ^{( 3 )} _{B} (a _{1,a} _{2,a} _{3,b} _{1,b} _{2,b} _{3,c;x} _{1,x} _{2,x} _{3)} = \sum ^{inf ty} _{i _{1,i} _{2,i} _{3=0}} ((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 ^{i _{1}} _{1} x ^{i _{2}} _{2} x ^{i _{3}} _{3}

for |x₁| < 1, |x₂| < 1, |x₃| < 1 and

F ^{( 3 )} _{C} (a,b,c _{1,c} _{2,c} _{3;x} _{1,x} _{2,x} _{3)} = \sum ^{inf ty} _{i _{1,i} _{2,i} _{3=0}} ((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 ^{i _{1}} _{1} x ^{i _{2}} _{2} x ^{i _{3}} _{3}

for |x₁|^1/2 + |x₂|^1/2 + |x₃|^1/2 < 1 and

F ^{( 3 )} _{D} (a,b _{1,b} _{2,b} _{3,c;x} _{1,x} _{2,x} _{3)} = \sum ^{inf ty} _{i _{1,i} _{2,i} _{3=0}} ((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 ^{i _{1}} _{1} x ^{i _{2}} _{2} x ^{i _{3}} _{3}

for |x₁| < 1, |x₂| < 1, |x₃| < 1. Here the Pochhammer symbol (q)_i indicates the i-th rising factorial of q, i.e.

(q) _{i} = q(q +1 ) … (q + i -1 ) = (\Gamma(q + i))/(\Gamma(q)) ~,

where the second equality is true for all complex

q except q =0 , -1 , -2 ,\ldots . These functions can be extended to other values of the variables x₁, x₂, x₃ by means of analytic continuation.

Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named F_E, F_F, ..., F_T and studied by Shanti 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 ^{(n)} _{A} (a, b _{1,\ldots,b} _{n,} c _{1,\ldots,c} _{n;} x _{1,\ldots,x} _{n)} = \sum ^{inf ty} _{i _{1,\ldots,i} _{n=0}} ((a) _{i _{1+ … +i} _{n}} (b _{1)} ^{i _{1}} … (b _{n)} ^{i _{n}})/((c _{1 )} ^{i _{1}} … (c _{n)} ^{i _{n}} i _{1!} … i _{n!}) x ^{i _{1}} _{1} … x ^{i _{n}} _{n} ~,

where |x₁| + ... + |x_n| < 1. These generalized series too are sometimes referred to as Lauricella functions.

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

F ^{( 2 )} _{A} \equiv F _{2} , F ^{( 2 )} _{B} \equiv F _{3} , F ^{( 2 )} _{C} \equiv F _{4} , F ^{( 2 )} _{D} \equiv F _{1.}

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

F ^{( 1 )} _{A} (a,b,c;x) \equiv F ^{( 1 )} _{B} (a,b,c;x) \equiv F ^{( 1 )} _{C} (a,b,c;x) \equiv F ^{( 1 )} _{D} (a,b,c;x) \equiv { _{2}F} _{1(a,b;c;x).}

Integral representation of F_D

In analogy with Appell's function F₁, Lauricella's F_D can be written as a one-dimensional Euler-type integral for any number n of variables:

F ^{(n)} _{D} (a, b _{1,\ldots,b} _{n,} c; x _{1,\ldots,x} _{n)} = (\Gamma(c))/(\Gamma(a) \Gamma(c - a)) \int ^{1} _{0} t ^{a-1} ( 1- t) ^{c-a-1} ( 1- x ^{- b _{1}} _{1 t)} … ( 1- x ^{- b _{n}} _{nt)} d t, \operatorname{Re} c > \operatorname{Re} a > 0 ~.

This representation can be easily verified by means of Taylor expansion of the integrand, followed by termwise integration. The representation implies that the incomplete elliptic integral Π is a special case of Lauricella's function F_D with three variables:

\Pi(n,\phi,k) = \int ^{\phi} _{0} (d \theta)/(( 1- n \sin ^{2} \theta) \sqrt{ 1- k ^{2} \sin ^{2} \theta) } = \sin (\phi) \,F_D^(\tfrac 1 2, 1, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; n \sin^2 \phi, \sin^2 \phi, k^2 \sin^2 \phi), \qquad |\operatorname \phi| < \frac ~.

Finite-sum solutions of F_D

Case 1 :

a>c , a - c a positive integer

One can relate F_D to the Carlson R function

R _{n} viaF _{D(a,\overline{b},c,\overline{z})=R} _{a-c} (\overline{b ^{*},} \overline{z ^{*})} ⋅ \prod _{i} (z ^{*)} _{i} ^{b ^{*} _{i}} = (\Gamma(a - c +1 )\Gamma(b ^{*)})/(\Gamma(a - c + b ^{*)}) ⋅ D _{a-c} (\overline{b ^{*},} \overline{z ^{*})} ⋅ \prod _{i} (z ^{*)} _{i} ^{b ^{*} _{i}}

with the iterative sum

D ^{*},} _{n(\overline{b} \overline{z ^{*}) = (1)/(n)} \sum ^{n} _{k =1} \left(\sum ^{N} _{i =1} b ^{*} _{i} ⋅ (z ^{*)} _{i} ^{k\right)} ⋅ D _{k-i} and D _{0=1}

where it can be exploited that the Carlson R function with

n> 0 has an exact representation (see ^[1] for more information).

The vectors are defined as

\overline{b ^{*}=[\overline{b},} c - \sum _{i} b _{i]} \overline{z ^{*} = [ (1)/(1- z _{1})} , \ldots, (1)/(1- z _{N-1}) , 1 ]

where the length of

\overline{z} and \overline{b} is N -1 , while the vectors \overline{z ^{*}} and \overline{b ^{*}} have length N .

Case 2:

c>a , c - a a positive integer

In this case there is also a known analytic form, but it is rather complicated to write down and involves several steps.See ^[2] for more information.

References

• Book: Appell . Paul . Paul Émile Appell . Kampé de Fériet . Joseph . Joseph Kampé de Fériet . Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite . French . Paris . Gauthier–Villars . 1926 . 52.0361.13 . (see p. 114)
• Book: Exton, Harold . Multiple hypergeometric functions and applications . Chichester, UK . Halsted Press, Ellis Horwood Ltd. . 1976 . Mathematics and its applications . 0-470-15190-0 . 0422713 .
• Lauricella . Giuseppe . Giuseppe Lauricella . Sulle funzioni ipergeometriche a più variabili . Italian . . 1893 . 7 . S1 . 111 - 158 . 10.1007/BF03012437 . 25.0756.01 . 122316343 .
• Saran . Shanti . Hypergeometric Functions of Three Variables . Ganita . 1954 . 5 . 1 . 77 - 91 . 0046-5402 . 0087777 . 0058.29602 . (corrigendum 1956 in Ganita 7, p. 65)
• Book: Slater, Lucy Joan . Lucy Joan Slater

. Lucy Joan Slater . Generalized hypergeometric functions . registration . Cambridge, UK . Cambridge University Press . 1966 . 0-521-06483-X . 0201688 . (there is a 2008 paperback with)

• Book: Srivastava . Hari M. . Karlsson . Per W. . Multiple Gaussian hypergeometric series . Chichester, UK . Halsted Press, Ellis Horwood Ltd. . 1985 . Mathematics and its applications . 0-470-20100-2 . 0834385 . (there is another edition with)

Notes and References

1. Glüsenkamp . T.. 2018. Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data . EPJ Plus . 133 . 6. 218. 10.1140/epjp/i2018-12042-x . 1712.01293. 2018EPJP..133..218G. 125665629.
2. Tan . J.. Zhou. P.. 2005. On the finite sum representations of the Lauricella functions FD . Advances in Computational Mathematics. 23 . 4. 333–351. 10.1007/s10444-004-1838-0. 7515235.