Appell series

For generalizations of Lambert series see Appell–Lerch series.

In mathematics, Appell series are a set of four hypergeometric series F₁, F₂, F₃, F₄ of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series ₂F₁ 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 F₁ is defined for |x| < 1, |y| < 1 by the double series

$F_1(a,b_1,b_2;c;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b_1{)}_m(b_2{)}_n}{(c{)}_{m+n}m!n!}{x}^m{y}^n,$

where $(q{)}_n$ is the rising factorial Pochhammer symbol. For other values of x and y the function F₁ can be defined by analytic continuation. It can be shown that

$F_1(a,b_1,b_2;c;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a{)}_r(b_1{)}_r(b_2{)}_r(c-a{)}_r}{(c+r-1{)}_r(c{)}_{2r}r!}{x}^r{y}^r{}_2{F}_1(a+r,b_1+r;c+2r;x){}_2{F}_1(a+r,b_2+r;c+2r;y).$

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

$F_2(a,b_1,b_2;c_1,c_2;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b_1{)}_m(b_2{)}_n}{(c_1{)}_m(c_2{)}_{n}m!n!}{x}^m{y}^n$

and it can be shown that

$F_2(a,b_1,b_2;c_1,c_2;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a{)}_r(b_1{)}_r(b_2{)}_r}{(c_1{)}_r(c_2{)}_{r}r!}{x}^r{y}^r{}_2{F}_1(a+r,b_1+r;c_1+r;x){}_2{F}_1(a+r,b_2+r;c_2+r;y).$

Also the function F₃ for |x| < 1, |y| < 1 can be defined by the series

$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a_1{)}_m(a_2{)}_n(b_1{)}_m(b_2{)}_n}{(c{)}_{m+n}m!n!}{x}^m{y}^n,$

and the function F₄ for |x|^1⁄2 + |y|^1⁄2 < 1 by the series

$F_4(a,b;c_1,c_2;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{(a{)}_{m+n}(b{)}_{m+n}}{(c_1{)}_m(c_2{)}_{n}m!n!}{x}^m{y}^n.$

The four series can each be represented as a simple series where the elements are Gaussian hypergeometric functions in terms of y:

$F_1(a,b_1,b_2;c;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a{)}_r(b_1{)}_r}{(c{)}_{r}r!}{x}^r{}_2{F}_1(a+r,b_2,c+r,y),$

$F_2(a,b_1,b_2;c_1,c_2;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a{)}_r(b_1{)}_r}{(c_1{)}_{r}r!}{x}^r{}_2{F}_1(a+r,b_2,c_2,y),$

$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a_1{)}_r(b_1{)}_r}{(c{)}_{r}r!}{x}^r{}_2{F}_1(a_2,b_2,c+r,y),$

$F_4(a,b;c_1,c_2;x,y)=\sum _{r=0}^{\mathrm{\infty }}\frac{(a{)}_r(b{)}_r}{(c_1{)}_{r}r!}{x}^r{}_2{F}_1(a+r,b+r,c_2,y).$

Similar expressions can be obtained upon exchange of the variables x and y and their respective parameters, e.g., c₁ and c₂ for F₄.

Recurrence relations

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:

$(a-b_1-b_2)F_1(a,b_1,b_2,c;x,y)-a{F}_1(a+1,b_1,b_2,c;x,y)+b_1{F}_1(a,b_1+1,b_2,c;x,y)+b_2{F}_1(a,b_1,b_2+1,c;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)-(c-a)F_1(a,b_1,b_2,c+1;x,y)-a{F}_1(a+1,b_1,b_2,c+1;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)+c(x-1)F_1(a,b_1+1,b_2,c;x,y)-(c-a)x{F}_1(a,b_1+1,b_2,c+1;x,y)=0,$

$c{F}_1(a,b_1,b_2,c;x,y)+c(y-1)F_1(a,b_1,b_2+1,c;x,y)-(c-a)y{F}_1(a,b_1,b_2+1,c+1;x,y)=0.$

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

Similarly, all recurrence relations for Appell's F₃ 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 F₁, 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 F₁ is further found to satisfy the following system of second-order differential equations:

$x(1-x)\frac{{\partial }^2{F}_1(x,y)}{\partial x^2}+y(1-x)\frac{{\partial }^2{F}_1(x,y)}{\partial x\partial y}+[c-(a+b_1+1)x]\frac{\partial F_1(x,y)}{\partial x}-b_{1}y\frac{\partial F_1(x,y)}{\partial y}-a{b}_1{F}_1(x,y)=0$

$y(1-y)\frac{{\partial }^2{F}_1(x,y)}{\partial y^2}+x(1-y)\frac{{\partial }^2{F}_1(x,y)}{\partial x\partial y}+[c-(a+b_2+1)y]\frac{\partial F_1(x,y)}{\partial y}-b_{2}x\frac{\partial F_1(x,y)}{\partial x}-a{b}_2{F}_1(x,y)=0$

A system partial differential equations for F₂ is

$x(1-x)\frac{{\partial }^2{F}_2(x,y)}{\partial x^2}-xy\frac{{\partial }^2{F}_2(x,y)}{\partial x\partial y}+[c_1-(a+b_1+1)x]\frac{\partial F_2(x,y)}{\partial x}-b_{1}y\frac{\partial F_2(x,y)}{\partial y}-a{b}_1{F}_2(x,y)=0$

$y(1-y)\frac{{\partial }^2{F}_2(x,y)}{\partial y^2}-xy\frac{{\partial }^2{F}_2(x,y)}{\partial x\partial y}+[c_2-(a+b_2+1)y]\frac{\partial F_2(x,y)}{\partial y}-b_{2}x\frac{\partial F_2(x,y)}{\partial x}-a{b}_2{F}_2(x,y)=0$

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 F₃ 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 F₃ 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 F₄ is

$x(1-x)\frac{{\partial }^2{F}_4(x,y)}{\partial x^2}-y^2\frac{{\partial }^2{F}_4(x,y)}{\partial y^2}-2xy\frac{{\partial }^2{F}_4(x,y)}{\partial x\partial y}+[c_1-(a+b+1)x]\frac{\partial F_4(x,y)}{\partial x}-(a+b+1)y\frac{\partial F_4(x,y)}{\partial y}-ab{F}_4(x,y)=0$

$y(1-y)\frac{{\partial }^2{F}_4(x,y)}{\partial y^2}-x^2\frac{{\partial }^2{F}_4(x,y)}{\partial x^2}-2xy\frac{{\partial }^2{F}_4(x,y)}{\partial x\partial y}+[c_2-(a+b+1)y]\frac{\partial F_4(x,y)}{\partial y}-(a+b+1)x\frac{\partial F_4(x,y)}{\partial x}-ab{F}_4(x,y)=0$

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 F₁ can also be written as a one-dimensional Euler-type integral:

$F_1(a,b_1,b_2,c;x,y)=\frac{\Gamma (c)}{\Gamma (a)\Gamma (c-a)}\underset{0}{\overset{1}{\int }}{t}^{a-1}(1-t{)}^{c-a-1}(1-xt{)}^{-b_1}(1-yt{)}^{-b_2}\mathrm{d}t,\mathrm{\Re }c>\mathrm{\Re }a>0.$

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 F₁:

$F(\varphi ,k)=\underset{0}{\overset{\varphi }{\int }}\frac{\mathrm{d}\theta }{\sqrt{1-k^2{\sin }^2\theta }}=\sin (\varphi )F_1(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),|\mathrm{\Re }\varphi |<\frac{\pi }{2},$

$E(\varphi ,k)=\underset{0}{\overset{\varphi }{\int }}\sqrt{1-k^2{\sin }^2\theta }\mathrm{d}\theta =\sin (\varphi )F_1(\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),|\mathrm{\Re }\varphi |<\frac{\pi }{2},$

$\Pi (n,k)=\underset{0}{\overset{\pi /2}{\int }}\frac{\mathrm{d}\theta }{(1-n{\sin }^2\theta )\sqrt{1-k^2{\sin }^2\theta }}=\frac{\pi }{2}{F}_1(\frac{1}{2},1,\frac{1}{2},1;n,k^2).$

Related series

Main articles: Humbert series and Lauricella hypergeometric 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.
• 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.

References

• (see also "Sur la série F₃(α,α',β,β',γ; x,y)" in C. R. Acad. Sci. 90, pp. 977–980)
• (see p. 14)
• .
• (see p. 224)
• (see also C. R. Acad. Sci. 90 (1880), pp. 1119–1121 and 1267–1269)
• (there is a 2008 paperback with )