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 ) = ∑ 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 1 i 1 x 2 i 2 x 3 i 3 {\displaystyle 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 |x 1 | + |x 2 | + |x 3 | < 1 and
F B ( 3 ) ( a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , c ; x 1 , x 2 , x 3 ) = ∑ 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 1 i 1 x 2 i 2 x 3 i 3 {\displaystyle 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 |x 1 | < 1, |x 2 | < 1, |x 3 | < 1 and
F C ( 3 ) ( a , b , c 1 , c 2 , c 3 ; x 1 , x 2 , x 3 ) = ∑ 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 1 i 1 x 2 i 2 x 3 i 3 {\displaystyle 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 |x 1 |½ + |x 2 |½ + |x 3 |½ < 1 and
F D ( 3 ) ( a , b 1 , b 2 , b 3 , c ; x 1 , x 2 , x 3 ) = ∑ 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 1 i 1 x 2 i 2 x 3 i 3 {\displaystyle 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 |x 1 | < 1, |x 2 | < 1, |x 3 | < 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 ) = Γ ( q + i ) Γ ( q ) , {\displaystyle (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 {\displaystyle q} except q = 0 , − 1 , − 2 , … {\displaystyle q=0,-1,-2,\ldots } .
These functions can be extended to other values of the variables x 1 , x 2 , x 3 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 [ edit ] These functions can be straightforwardly extended to n variables. One writes for example
F A ( n ) ( a , b 1 , … , b n , c 1 , … , c n ; x 1 , … , x n ) = ∑ i 1 , … , 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 1 i 1 ⋯ x n i n , {\displaystyle 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 |x 1 | + ... + |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 ) ≡ F 2 , F B ( 2 ) ≡ F 3 , F C ( 2 ) ≡ F 4 , F D ( 2 ) ≡ F 1 . {\displaystyle 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 ) ≡ F B ( 1 ) ( a , b , c ; x ) ≡ F C ( 1 ) ( a , b , c ; x ) ≡ F D ( 1 ) ( a , b , c ; x ) ≡ 2 F 1 ( a , b ; c ; x ) . {\displaystyle 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 [ edit ] In analogy with Appell's function F 1 , 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 , … , b n , c ; x 1 , … , x n ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − x 1 t ) − b 1 ⋯ ( 1 − x n t ) − b n d t , Re c > Re a > 0 . {\displaystyle 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:
Π ( n , ϕ , k ) = ∫ 0 ϕ d θ ( 1 − n sin 2 θ ) 1 − k 2 sin 2 θ = sin ( ϕ ) F D ( 3 ) ( 1 2 , 1 , 1 2 , 1 2 , 3 2 ; n sin 2 ϕ , sin 2 ϕ , k 2 sin 2 ϕ ) , | Re ϕ | < π 2 . {\displaystyle \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 [ edit ] Case 1 : a > c {\displaystyle a>c} , a − c {\displaystyle a-c} a positive integer
One can relate F D to the Carlson R function R n {\displaystyle R_{n}} via
F D ( a , b ¯ , c , z ¯ ) = R a − c ( b ∗ ¯ , z ∗ ¯ ) ⋅ ∏ i ( z i ∗ ) b i ∗ = Γ ( a − c + 1 ) Γ ( b ∗ ) Γ ( a − c + b ∗ ) ⋅ D a − c ( b ∗ ¯ , z ∗ ¯ ) ⋅ ∏ i ( z i ∗ ) b i ∗ {\displaystyle 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 ( b ∗ ¯ , z ∗ ¯ ) = 1 n ∑ k = 1 n ( ∑ i = 1 N b i ∗ ⋅ ( z i ∗ ) k ) ⋅ D k − i {\displaystyle 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 {\displaystyle D_{0}=1}
where it can be exploited that the Carlson R function with n > 0 {\displaystyle n>0} has an exact representation (see [1] for more information).
The vectors are defined as
b ∗ ¯ = [ b ¯ , c − ∑ i b i ] {\displaystyle {\overline {b^{*}}}=[{\overline {b}},c-\sum _{i}b_{i}]}
z ∗ ¯ = [ 1 1 − z 1 , … , 1 1 − z N − 1 , 1 ] {\displaystyle {\overline {z^{*}}}=[{\frac {1}{1-z_{1}}},\ldots ,{\frac {1}{1-z_{N-1}}},1]}
where the length of z ¯ {\displaystyle {\overline {z}}} and b ¯ {\displaystyle {\overline {b}}} is N − 1 {\displaystyle N-1} , while the vectors z ∗ ¯ {\displaystyle {\overline {z^{*}}}} and b ∗ ¯ {\displaystyle {\overline {b^{*}}}} have length N {\displaystyle N} .
Case 2: c > a {\displaystyle c>a} , c − a {\displaystyle 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 [ edit ] 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 )