Appell series

The Appell series F₁ is defined for |x| < 1, |y| < 1 by the double series:

${\displaystyle F_{1}(a,b_{1},b_{2};c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,}$

${\displaystyle F_{1}(a,b_{1},b_{2};c;x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}(c-a)_{r}}{(c+r-1)_{r}(c)_{2r}r!}}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c+2r;x\right){}_{2}F_{1}\left(a+r,b_{2}+r;c+2r;y\right)~}$

where the Pochhammer symbol (q)_n represents the rising factorial:

${\displaystyle (q)_{n}=q\,(q+1)\cdots (q+n-1)={\frac {\Gamma (q+n)}{\Gamma (q)}}~,}$

where the second equality is true for all complex ${\displaystyle q}$ except ${\displaystyle q=0,-1,-2,\ldots }$.

For other values of x and y the function F₁ can be defined by analytic continuation.

Similarly, the function F₂ is defined for |x| + |y| < 1 by the series:

${\displaystyle F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,}$

${\displaystyle F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}}{(c_{1})_{r}(c_{2})_{r}r!}}\,x^{r}y^{r}{}_{2}F_{1}\left(a+r,b_{1}+r;c_{1}+2r;x\right){}_{2}F_{1}\left(a+r,b_{2}+r;c_{2}+2r;y\right)~}$

the function F₃ for |x| < 1, |y| < 1 by the series:

${\displaystyle F_{3}(a_{1},a_{2},b_{1},b_{2};c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,}$

and the function F₄ for |x|^½ + |y|^½ < 1 by the series:

${\displaystyle F_{4}(a,b;c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~.}$

Like the Gauss hypergeometric series ₂F₁, the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's F₁ is given by:

${\displaystyle (a-b_{1}-b_{2})F_{1}(a,b_{1},b_{2},c;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c;x,y)+b_{1}F_{1}(a,b_{1}+1,b_{2},c;x,y)+b_{2}F_{1}(a,b_{1},b_{2}+1,c;x,y)=0~,}$

${\displaystyle c\,F_{1}(a,b_{1},b_{2},c;x,y)-(c-a)F_{1}(a,b_{1},b_{2},c+1;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c+1;x,y)=0~,}$

${\displaystyle c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(x-1)F_{1}(a,b_{1}+1,b_{2},c;x,y)-(c-a)x\,F_{1}(a,b_{1}+1,b_{2},c+1;x,y)=0~,}$

${\displaystyle c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(y-1)F_{1}(a,b_{1},b_{2}+1,c;x,y)-(c-a)y\,F_{1}(a,b_{1},b_{2}+1,c+1;x,y)=0~.}$

Any other relation[1] valid for F₁ can be derived from these four.

Similarly, all recurrence relations for Appell's F₃ follow from this set of five:

${\displaystyle c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)+(a_{1}+a_{2}-c)F_{3}(a_{1},a_{2},b_{1},b_{2},c+1;x,y)-a_{1}F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c+1;x,y)-a_{2}F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c+1;x,y)=0~,}$

${\displaystyle c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c;x,y)+b_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,}$

${\displaystyle c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c;x,y)+b_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~,}$

${\displaystyle c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1}+1,b_{2},c;x,y)+a_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,}$

${\displaystyle c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1},b_{2}+1,c;x,y)+a_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~.}$

For Appell's F₁, the following derivatives result from the definition by a double series:

${\displaystyle {\frac {\partial ^{n}}{\partial x^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{1}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1}+n,b_{2},c+n;x,y)}$

${\displaystyle {\frac {\partial ^{n}}{\partial y^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{2}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1},b_{2}+n,c+n;x,y)}$

From its definition, Appell's F₁ is further found to satisfy the following system of second-order differential equations:

${\displaystyle x(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial F_{1}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{1}(x,y)}{\partial y}}-ab_{1}F_{1}(x,y)=0}$

${\displaystyle y(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial F_{1}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{1}(x,y)}{\partial x}}-ab_{2}F_{1}(x,y)=0}$

A system partial differential equations for F₂ is

${\displaystyle x(1-x){\frac {\partial ^{2}F_{2}(x,y)}{\partial x^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b_{1}+1)x]{\frac {\partial F_{2}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{2}(x,y)}{\partial y}}-ab_{1}F_{2}(x,y)=0}$

${\displaystyle y(1-y){\frac {\partial ^{2}F_{2}(x,y)}{\partial y^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b_{2}+1)x]{\frac {\partial F_{2}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{2}(x,y)}{\partial x}}-ab_{2}F_{2}(x,y)=0}$

The system have solution

${\displaystyle F_{2}(x,y)=C_{1}F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{2}(a-c_{1}+1,b_{1}-c_{1}+1,b_{2},2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{2}(a-c_{2}+1,b_{1},b_{2}-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{2}(a-c_{1}-c_{2}+2,b_{1}-c_{1}+1,b_{2}-c_{2}+1,2-c_{1},2-c_{2};x,y)}$

Similarly, for F₃ the following derivatives result from the definition:

${\displaystyle {\frac {\partial }{\partial x}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{1}b_{1}}{c}}F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)}$

${\displaystyle {\frac {\partial }{\partial y}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{2}b_{2}}{c}}F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)}$

And for F₃ the following system of differential equations is obtained:

${\displaystyle x(1-x){\frac {\partial ^{2}F_{3}(x,y)}{\partial x^{2}}}+y{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{1}+b_{1}+1)x]{\frac {\partial F_{3}(x,y)}{\partial x}}-a_{1}b_{1}F_{3}(x,y)=0}$

${\displaystyle y(1-y){\frac {\partial ^{2}F_{3}(x,y)}{\partial y^{2}}}+x{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{2}+b_{2}+1)y]{\frac {\partial F_{3}(x,y)}{\partial y}}-a_{2}b_{2}F_{3}(x,y)=0}$

A system partial differential equations for F₄ is

${\displaystyle x(1-x){\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-y^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b+1)x]{\frac {\partial F_{4}(x,y)}{\partial x}}-(a+b+1)y{\frac {\partial F_{4}(x,y)}{\partial y}}-abF_{4}(x,y)=0}$

${\displaystyle y(1-y){\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-x^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b+1)y]{\frac {\partial F_{4}(x,y)}{\partial y}}-(a+b+1)x{\frac {\partial F_{4}(x,y)}{\partial x}}-abF_{4}(x,y)=0}$

The system have solution

${\displaystyle F_{4}(x,y)=C_{1}F_{4}(a,b,c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{4}(a-c_{1}+1,b-c_{1}+1,2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{4}(a-c_{2}+1,b-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{4}(2+a-c_{1}-c_{2},2+b-c_{1}-c_{2},2-c_{1},2-c_{2};x,y)}$

The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn & Ryzhik 2015, §9.184) harv error: no target: CITEREFGradshteynRyzhik2015 (help). However, Émile Picard (1881) discovered that Appell's F₁ can also be written as a one-dimensional Euler-type integral:

${\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.}$

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

Picard's integral representation implies that the incomplete elliptic integrals F and E as well as the complete elliptic integral Π are special cases of Appell's F₁:

${\displaystyle F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\sin(\phi )\,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,}$

${\displaystyle E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\mathrm {d} \theta =\sin(\phi )\,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},-{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,}$

${\displaystyle \Pi (n,k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}={\frac {\pi }{2}}\,F_{1}({\tfrac {1}{2}},1,{\tfrac {1}{2}},1;n,k^{2})~.}$

• Main article: Humbert series

There are seven related series of two variables, Φ₁, Φ₂, Φ₃, Ψ₁, Ψ₂, Ξ₁, and Ξ₂, which generalize Kummer's confluent hypergeometric function ₁F₁ of one variable and the confluent hypergeometric limit function ₀F₁ of one variable in a similar manner. The first of these was introduced by Pierre Humbert in 1920.

• Main article: Lauricella hypergeometric series

Giuseppe Lauricella (1893) defined four functions similar to the Appell series, but depending on many variables rather than just the two variables x and y. These series were also studied by Appell. They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and contour integrals.

• Appell, Paul (1880). "Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 90: 296–298 and 731–735. JFM 12.0296.01.CS1 maint: ref=harv (link) (see also "Sur la série F₃(α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90, pp. 977–980)
• Appell, Paul (1882). "Sur les fonctions hypergéométriques de deux variables". Journal de Mathématiques Pures et Appliquées. (3ème série) (in French). 8: 173–216.CS1 maint: ref=harv (link)
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• Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF). New York: McGraw–Hill.CS1 maint: ref=harv (link) (see p. 224)
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "9.18.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.CS1 maint: ref=harv (link)
• Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 171: 490–492. JFM 47.0348.01.CS1 maint: ref=harv (link)
• Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo (in Italian). 7: 111–158. doi:10.1007/BF03012437. JFM 25.0756.01.CS1 maint: ref=harv (link)
• Picard, Émile (1881). "Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques". Annales Scientifiques de l'École Normale Supérieure. Série 2 (in French). 10: 305–322. JFM 13.0389.01.CS1 maint: ref=harv (link) (see also C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269)
• Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688.CS1 maint: ref=harv (link) (there is a 2008 paperback with ISBN 978-0-521-09061-2)

• Aarts, Ronald M. "Lauricella Functions". MathWorld.
• Weisstein, Eric W. "Appell Hypergeometric Function". MathWorld.