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| In 1893 [[Giuseppe Lauricella]] defined and studied four [[hypergeometric series]] ''F''<sub>''A''</sub>, ''F''<sub>''B''</sub>, ''F''<sub>''C''</sub>, ''F''<sub>''D''</sub> of three variables. They are {{harv|Lauricella|1893}}:
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| :<math>
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| F_A^{(3)}(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) =
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| \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}
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| </math>
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| for |''x''<sub>1</sub>| + |''x''<sub>2</sub>| + |''x''<sub>3</sub>| < 1 and
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| :<math>
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| F_B^{(3)}(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3) =
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| \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}
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| </math>
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| for |''x''<sub>1</sub>| < 1, |''x''<sub>2</sub>| < 1, |''x''<sub>3</sub>| < 1 and
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| :<math>
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| F_C^{(3)}(a,b,c_1,c_2,c_3;x_1,x_2,x_3) =
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| \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}
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| </math>
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| for |''x''<sub>1</sub>|<sup>½</sup> + |''x''<sub>2</sub>|<sup>½</sup> + |''x''<sub>3</sub>|<sup>½</sup> < 1 and
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| :<math>
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| F_D^{(3)}(a,b_1,b_2,b_3,c;x_1,x_2,x_3) =
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| \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}
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| </math>
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| for |''x''<sub>1</sub>| < 1, |''x''<sub>2</sub>| < 1, |''x''<sub>3</sub>| < 1. Here the [[Pochhammer symbol]] (''q'')<sub>''i''</sub> indicates the ''i''-th rising factorial power of ''q'', i.e.
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| :<math>
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| (q)_{i} = \frac{\Gamma(q+i)} {\Gamma(q)} = q\,(q+1) \cdots (q+i-1).
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| </math>
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|
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| These functions can be extended to other values of the variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub> by means of [[analytic continuation]].
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| Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named ''F''<sub>''E''</sub>, ''F''<sub>''F''</sub>, ..., ''F''<sub>''T''</sub> and studied by Shanti Saran in 1954 {{harv|Saran|1954}}. There are therefore a total of 14 Lauricella–Saran hypergeometric functions.
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| ==Generalization to ''n'' variables==
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| These functions can be straightforwardly extended to ''n'' variables. One writes for example
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| :<math>
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| F_A^{(n)}(a, b_1,\ldots,b_n, c_1,\ldots,c_n; x_1,\ldots,x_n) =
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| \sum_{i_1,\ldots,i_n=0}^{\infty} \frac{(a)_{i_1+\ldots+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} ~,
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| </math>
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| where |''x''<sub>1</sub>| + ... + |''x''<sub>''n''</sub>| < 1. These generalized series too are sometimes referred to as Lauricella functions.
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| When ''n'' = 2, the Lauricella functions correspond to the [[Appell series|Appell hypergeometric series]] of two variables:
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| :<math>
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| 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.
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| </math>
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| When ''n'' = 1, all four functions reduce to the [[hypergeometric function|Gauss hypergeometric function]]:
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| :<math>
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| 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).
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| </math>
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| ==Integral representation of ''F''<sub>''D''</sub>==
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| In analogy with [[Appell series|Appell's function ''F''<sub>1</sub>]], Lauricella's ''F''<sub>''D''</sub> can be written as a one-dimensional [[Leonhard Euler|Euler]]-type [[integral]] for any number ''n'' of variables:
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| :<math> | |
| F_D^{(n)}(a, b_1,\ldots,b_n, c; x_1,\ldots,x_n) =
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| \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-x_1t)^{-b_1} \cdots (1-x_nt)^{-b_n} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~.
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| </math>
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| This representation can be easily verified by means of [[Taylor series|Taylor expansion]] of the integrand, followed by termwise integration. The representation implies that the [[elliptic integral|incomplete elliptic integral]] Π is a special case of Lauricella's function ''F''<sub>''D''</sub> with three variables:
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| :<math>
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| \Pi(n,\phi,k) =
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| \int_0^{\phi} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta) \sqrt{1 - k^2 \sin^2 \theta}} =
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| \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), \quad |\real \,\phi| < \frac \pi 2 ~.
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| </math>
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| ==References==
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| * {{cite book | last1= Appell | first1= Paul | author1-link= Paul Émile Appell | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= French | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 | ref= harv}} (see p. 114)
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| * {{cite book | last= Exton | first= Harold | title= Multiple hypergeometric functions and applications | location= Chichester, UK | publisher= Halsted Press, Ellis Horwood Ltd. | year= 1976 | series= Mathematics and its applications | isbn= 0-470-15190-0 | mr= 0422713 | ref= harv}}
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| * {{cite journal | last= Lauricella | first= Giuseppe | authorlink= Giuseppe Lauricella | title= Sulle funzioni ipergeometriche a più variabili | language= Italian | journal= [[Rendiconti del Circolo Matematico di Palermo]] | year= 1893 | volume= 7 | issue= S1 | pages= 111–158 | doi= 10.1007/BF03012437 | jfm= 25.0756.01 | ref= harv}}
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| * {{cite journal | last= Saran | first= Shanti | title= Hypergeometric Functions of Three Variables | journal= Ganita | year= 1954 | volume= 5 | issue= 1 | pages= 77–91 | issn= 0046-5402 | mr= 0087777 | zbl= 0058.29602 | ref= harv}} (corrigendum 1956 in ''Ganita'' '''7''', p. 65)
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| * {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Generalized hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | ref= harv}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)
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| * {{cite book | last1= Srivastava | first1= Hari M. | last2= Karlsson | first2= Per W. | title= Multiple Gaussian hypergeometric series | location= Chichester, UK | publisher= Halsted Press, Ellis Horwood Ltd. | year= 1985 | series= Mathematics and its applications | isbn= 0-470-20100-2 | mr= 0834385 | ref= harv}} (there is another edition with ISBN 0-85312-602-X)
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| ==External links==
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| * {{mathworld | urlname= LauricellaFunctions | title= Lauricella Functions | author= Ronald M. Aarts}}
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| {{DEFAULTSORT:Lauricella Hypergeometric Series}}
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| [[Category:Hypergeometric functions]]
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| [[Category:Mathematical series]]
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Golda is what's created on my birth certification although it is not the name on my beginning certification. It's not a common thing but what I like performing is to climb but I don't have the time recently. Ohio is where my house is but my husband wants us to transfer. He works as a bookkeeper.
Feel free to surf to my web site ... psychics online