# Appell series

In mathematics, Appell series are a set of four hypergeometric series F$_1$, F$_2$, F$_3$, F$_4$ of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series $_2$F$_1$ of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.

## Definitions

The Appell series F$_1$ is defined for |x| < 1, |y| < 1 by the double series
    $F_{1}(a,b_{1},b_{2};c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$
where $(q)_{n}$ is the Pochhammer symbol. For other values of x and y the function F$_1$ can be defined by analytic continuation. It can be shown that
    $F_{1}(a,b_{1},b_{2};c;x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}(c-a)_{r}}{(c+r-1)_{r}(c)_{2r}r!}}\,x^{r}y^{r}_{2}F_{1}\left(a+r,b_{1}+r;c+2r;x\right)_{2}F_{1}\left(a+r,b_{2}+r;c+2r;y\right)~.$
Similarly, the function F$_2$ is defined for |x| + |y| < 1 by the series
    $F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}$
and it can be shown that
    $F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum _{r=0}^{\infty }{\frac {(a)_{r}(b_{1})_{r}(b_{2})_{r}}{(c_{1})_{r}(c_{2})_{r}r!}}\,x^{r}y^{r}_{2}F_{1}\left(a+r,b_{1}+r;c_{1}+r;x\right)_{2}F_{1}\left(a+r,b_{2}+r;c_{2}+r;y\right)~.$
Also the function F$_3$ for |x| < 1, |y| < 1 can be defined by the series
    $F_{3}(a_{1},a_{2},b_{1},b_{2};c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$
and the function F$_4$ for |x|$^{1⁄2}$ + |y|$^{1⁄2}$ < 1 by the series
    $F_{4}(a,b;c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~.$

## Recurrence relations

Like the Gauss hypergeometric series $_2$F$_1$, the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's F$_1$ is given by:
    $(a-b_{1}-b_{2})F_{1}(a,b_{1},b_{2},c;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c;x,y)+b_{1}F_{1}(a,b_{1}+1,b_{2},c;x,y)+b_{2}F_{1}(a,b_{1},b_{2}+1,c;x,y)=0~,$
    $c\,F_{1}(a,b_{1},b_{2},c;x,y)-(c-a)F_{1}(a,b_{1},b_{2},c+1;x,y)-a\,F_{1}(a+1,b_{1},b_{2},c+1;x,y)=0~,$
    $c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(x-1)F_{1}(a,b_{1}+1,b_{2},c;x,y)-(c-a)x\,F_{1}(a,b_{1}+1,b_{2},c+1;x,y)=0~,$
    $c\,F_{1}(a,b_{1},b_{2},c;x,y)+c(y-1)F_{1}(a,b_{1},b_{2}+1,c;x,y)-(c-a)y\,F_{1}(a,b_{1},b_{2}+1,c+1;x,y)=0~.$
Any other relation valid for F$_1$ can be derived from these four.
Similarly, all recurrence relations for Appell's F$_3$ follow from this set of five:
    $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)+(a_{1}+a_{2}-c)F_{3}(a_{1},a_{2},b_{1},b_{2},c+1;x,y)-a_{1}F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c+1;x,y)-a_{2}F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c+1;x,y)=0~,$
    $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1}+1,a_{2},b_{1},b_{2},c;x,y)+b_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,$
    $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2},c;x,y)+b_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~,$
    $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1}+1,b_{2},c;x,y)+a_{1}x\,F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)=0~,$
    $c\,F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)-c\,F_{3}(a_{1},a_{2},b_{1},b_{2}+1,c;x,y)+a_{2}y\,F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)=0~.$

## Derivatives and differential equations

For Appell's F$_1$, the following derivatives result from the definition by a double series:
    ${\frac {\partial ^{n}}{\partial x^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{1}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1}+n,b_{2},c+n;x,y)$
    ${\frac {\partial ^{n}}{\partial y^{n}}}F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\left(a\right)_{n}\left(b_{2}\right)_{n}}{\left(c\right)_{n}}}F_{1}(a+n,b_{1},b_{2}+n,c+n;x,y)$
From its definition, Appell's F$_1$ is further found to satisfy the following system of second-order differential equations:
    $x(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x^{2}}}+y(1-x){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{1}+1)x]{\frac {\partial F_{1}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{1}(x,y)}{\partial y}}-ab_{1}F_{1}(x,y)=0$
    $y(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial y^{2}}}+x(1-y){\frac {\partial ^{2}F_{1}(x,y)}{\partial x\partial y}}+[c-(a+b_{2}+1)y]{\frac {\partial F_{1}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{1}(x,y)}{\partial x}}-ab_{2}F_{1}(x,y)=0$
A system partial differential equations for F$_2$ is
    $x(1-x){\frac {\partial ^{2}F_{2}(x,y)}{\partial x^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b_{1}+1)x]{\frac {\partial F_{2}(x,y)}{\partial x}}-b_{1}y{\frac {\partial F_{2}(x,y)}{\partial y}}-ab_{1}F_{2}(x,y)=0$
    $y(1-y){\frac {\partial ^{2}F_{2}(x,y)}{\partial y^{2}}}-xy{\frac {\partial ^{2}F_{2}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b_{2}+1)y]{\frac {\partial F_{2}(x,y)}{\partial y}}-b_{2}x{\frac {\partial F_{2}(x,y)}{\partial x}}-ab_{2}F_{2}(x,y)=0$
The system have solution
    $F_{2}(x,y)=C_{1}F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{2}(a-c_{1}+1,b_{1}-c_{1}+1,b_{2},2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{2}(a-c_{2}+1,b_{1},b_{2}-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{2}(a-c_{1}-c_{2}+2,b_{1}-c_{1}+1,b_{2}-c_{2}+1,2-c_{1},2-c_{2};x,y)$
Similarly, for F$_3$ the following derivatives result from the definition:
    ${\frac {\partial }{\partial x}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{1}b_{1}}{c}}F_{3}(a_{1}+1,a_{2},b_{1}+1,b_{2},c+1;x,y)$
    ${\frac {\partial }{\partial y}}F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)={\frac {a_{2}b_{2}}{c}}F_{3}(a_{1},a_{2}+1,b_{1},b_{2}+1,c+1;x,y)$
And for F$_3$ the following system of differential equations is obtained:
    $x(1-x){\frac {\partial ^{2}F_{3}(x,y)}{\partial x^{2}}}+y{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{1}+b_{1}+1)x]{\frac {\partial F_{3}(x,y)}{\partial x}}-a_{1}b_{1}F_{3}(x,y)=0$
    $y(1-y){\frac {\partial ^{2}F_{3}(x,y)}{\partial y^{2}}}+x{\frac {\partial ^{2}F_{3}(x,y)}{\partial x\partial y}}+[c-(a_{2}+b_{2}+1)y]{\frac {\partial F_{3}(x,y)}{\partial y}}-a_{2}b_{2}F_{3}(x,y)=0$
A system partial differential equations for F$_4$ is
    $x(1-x){\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-y^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{1}-(a+b+1)x]{\frac {\partial F_{4}(x,y)}{\partial x}}-(a+b+1)y{\frac {\partial F_{4}(x,y)}{\partial y}}-abF_{4}(x,y)=0$
    $y(1-y){\frac {\partial ^{2}F_{4}(x,y)}{\partial y^{2}}}-x^{2}{\frac {\partial ^{2}F_{4}(x,y)}{\partial x^{2}}}-2xy{\frac {\partial ^{2}F_{4}(x,y)}{\partial x\partial y}}+[c_{2}-(a+b+1)y]{\frac {\partial F_{4}(x,y)}{\partial y}}-(a+b+1)x{\frac {\partial F_{4}(x,y)}{\partial x}}-abF_{4}(x,y)=0$
The system has solution
    $F_{4}(x,y)=C_{1}F_{4}(a,b,c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{4}(a-c_{1}+1,b-c_{1}+1,2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{4}(a-c_{2}+1,b-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{4}(2+a-c_{1}-c_{2},2+b-c_{1}-c_{2},2-c_{1},2-c_{2};x,y)$

## Integral representations

The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn et al. 2015, §9.184). However, Émile Picard (1881) discovered that Appell's F$_1$ can also be written as a one-dimensional Euler-type integral:
    $F_{1}(a,b_{1},b_{2},c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.$
This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

## Special cases

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 F$_1$:
    $F(\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\sin(\phi )\,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,$
    $E(\phi ,k)=\int _{0}^{\phi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,\mathrm {d} \theta =\sin(\phi )\,F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}},-{\tfrac {1}{2}},{\tfrac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\quad |\Re \,\phi |<{\frac {\pi }{2}}~,$
    $\Pi (n,k)=\int _{0}^{\pi /2}{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}={\frac {\pi }{2}}\,F_{1}({\tfrac {1}{2}},1,{\tfrac {1}{2}},1;n,k^{2})~.$

## Related series

* There are seven related series of two variables, Φ$_1$, Φ$_2$, Φ$_3$, Ψ$_1$, Ψ$_2$, Ξ$_1$, and Ξ$_2$, which generalize Kummer's confluent hypergeometric function $_1$F$_1$ of one variable and the confluent hypergeometric limit function $_0$F$_1$ of one variable in a similar manner. The first of these was introduced by Pierre Humbert in 1920.
* 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.