Lauricella hypergeometric series

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

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}}

for |x₁| + |x₂| + |x₃| < 1 and

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}}

for |x₁| < 1, |x₂| < 1, |x₃| < 1 and

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}}

for |x₁|^½ + |x₂|^½ + |x₃|^½ < 1 and

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}}

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)\cdots (q+i-1)={\frac {\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 (Saran 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})=\sum _{i_{1},\ldots ,i_{n}=0}^{\infty }{\frac {(a)_{i_{1}+\cdots +i_{n}}(b_{1})_{i_{1}}\cdots (b_{n})_{i_{n}}}{(c_{1})_{i_{1}}\cdots (c_{n})_{i_{n}}\,i_{1}!\cdots \,i_{n}!}}\,x_{1}^{i_{1}}\cdots x_{n}^{i_{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_{A}^{(2)}\equiv F_{2},\quad F_{B}^{(2)}\equiv F_{3},\quad F_{C}^{(2)}\equiv F_{4},\quad F_{D}^{(2)}\equiv F_{1}.

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

F_{A}^{(1)}(a,b,c;x)\equiv F_{B}^{(1)}(a,b,c;x)\equiv F_{C}^{(1)}(a,b,c;x)\equiv F_{D}^{(1)}(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_{D}^{(n)}(a,b_{1},\ldots ,b_{n},c;x_{1},\ldots ,x_{n})={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-x_{1}t)^{-b_{1}}\cdots (1-x_{n}t)^{-b_{n}}\,\mathrm {d} t,\qquad \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 _{0}^{\phi }{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}=\sin(\phi )\,F_{D}^{(3)}({\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 {Re} \phi |<{\frac {\pi }{2}}~.

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 $R_{n}$ via

$F_{D}(a,{\overline {b}},c,{\overline {z}})=R_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}={\frac {\Gamma (a-c+1)\Gamma (b^{*})}{\Gamma (a-c+b^{*})}}\cdot D_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}$

with the iterative sum

$D_{n}({\overline {b^{*}}},{\overline {z^{*}}})={\frac {1}{n}}\sum _{k=1}^{n}\left(\sum _{i=1}^{N}b_{i}^{*}\cdot (z_{i}^{*})^{k}\right)\cdot D_{k-i}$ 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

${\overline {b^{*}}}=[{\overline {b}},c-\sum _{i}b_{i}]$

${\overline {z^{*}}}=[{\frac {1}{1-z_{1}}},\ldots ,{\frac {1}{1-z_{N-1}}},1]$

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

^ Glüsenkamp, T. (2018). "Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data". EPJ Plus. 133 (6): 218. arXiv:1712.01293. Bibcode:2018EPJP..133..218G. doi:10.1140/epjp/i2018-12042-x. S2CID 125665629.
^ Tan, J.; Zhou, P. (2005). "On the finite sum representations of the Lauricella functions FD". Advances in Computational Mathematics. 23 (4): 333–351. doi:10.1007/s10444-004-1838-0. S2CID 7515235.

Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (see p. 114)
Exton, Harold (1976). Multiple hypergeometric functions and applications. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. ISBN 0-470-15190-0. MR 0422713.
Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo (in Italian). 7 (S1): 111–158. doi:10.1007/BF03012437. JFM 25.0756.01. S2CID 122316343.
Saran, Shanti (1954). "Hypergeometric Functions of Three Variables". Ganita. 5 (1): 77–91. ISSN 0046-5402. MR 0087777. Zbl 0058.29602. (corrigendum 1956 in Ganita 7, p. 65)
Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)
Srivastava, Hari M.; Karlsson, Per W. (1985). Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. ISBN 0-470-20100-2. MR 0834385. (there is another edition with ISBN 0-85312-602-X)

Ronald M. Aarts. "Lauricella Functions". MathWorld.

Sequences and series

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced (list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)