Appell series

In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by 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}^\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 F1 can be defined by analytic continuation. It can be shown that


 * $$F_1(a,b_1,b_2;c;x,y) = \sum_{r=0}^\infty \frac{(a)_r (b_1)_r (b_2)_r (c-a)_r} {(c+r-1)_r (c)_{2r} r!} \,x^r y^r {}_2F_{1}\left(a+r,b_1+r;c+2r;x\right){}_2F_{1}\left(a+r,b_2+r;c+2r;y\right)~.$$

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}^\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 {}_2F_{1}\left(a+r,b_1+r;c_1+r;x\right){}_2F_{1}\left(a+r,b_2+r;c_2+r;y\right)~.$$ 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}^\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}^\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 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^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_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^2F_1(x,y)} {\partial y^2} + x(1-y) \frac {\partial^2F_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^2F_2(x,y)} {\partial x^2} - xy \frac {\partial^2F_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^2F_2(x,y)} {\partial y^2} - xy \frac {\partial^2F_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


 * $$F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)$$

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^2F_3(x,y)} {\partial x^2} + y \frac {\partial^2F_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^2F_3(x,y)} {\partial y^2} + x \frac {\partial^2F_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^2F_4(x,y)} {\partial x^2} - y^2 \frac {\partial^2F_4(x,y)} {\partial y^2} -2xy\frac {\partial^2F_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}-a b F_4(x,y)= 0 $$



y(1-y) \frac {\partial^2F_4(x,y)} {\partial y^2} - x^2 \frac {\partial^2F_4(x,y)} {\partial x^2} -2xy\frac {\partial^2F_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}-a b F_4(x,y)= 0 $$ The system has solution


 * $$F_4(x,y)=C_1F_4(a,b,c_1,c_2;x,y)+C_2x^{1-c_1}F_4(a-c_1+1,b-c_1+1,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_4(a-c_2+1,b-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_4(2+a-c_1-c_2,2+b-c_1-c_2,2-c_1,2-c_2;x,y)$$

Integral representations
The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only. However, 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)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \,\mathrm{d}t, \quad \real \,c > \real \,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(\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 |\real \,\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 |\real \,\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, 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.
 * 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.