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 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

:$F_1(a,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

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

:$F_1(a,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^r y^r _2F_\left(a+r,b_1+r;c+2r;x\right)_2F_\left(a+r,b_2+r;c+2r;y\right)~.$

Similarly, the function ''F''₂ is defined for , ''x'', + , ''y'', < 1 by the series

:$F_2(a,b_1,b_2;c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n$

and it can be shown that

:$F_2(a,b_1,b_2;c_1,c_2;x,y) = \sum_^\infty \frac \,x^r y^r _2F_\left(a+r,b_1+r;c_1+r;x\right)_2F_\left(a+r,b_2+r;c_2+r;y\right)~.$

Also the function ''F''₃ for , ''x'', < 1, , ''y'', < 1 can be defined by the series

:$F_3(a_1,a_2,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

and the function ''F''₄ for , ''x'', ^½ + , ''y'', ^½ < 1 by the series

:$F_4(a,b;c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~.$

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 relationFor example, $(y-x) F_1(a, b_1+1, b_2+1,c,x,y) = y \, F_1(a,b_1,b_2+1,c,x,y) - x \, F_1(a,b_1+1,b_2,c,x,y)$ 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:

:$\frac F_1(a,b_1,b_2,c; x,y) = \frac F_1(a+n,b_1+n,b_2,c+n; x,y)$

:$\frac F_1(a,b_1,b_2,c; x,y) = \frac F_1(a+n,b_1,b_2+n,c+n; x,y)$

From its definition, Appell's ''F''₁ is further found to satisfy the following system of second-order differential equations:

:
x(1-x) \frac + y(1-x) \frac
+ - (a+b_1+1) x\frac - b_1 y
\frac - a b_1 F_1(x,y) = 0

:
y(1-y) \frac + x(1-y) \frac
+ - (a+b_2+1) y\frac - b_2 x
\frac - a b_2 F_1(x,y)= 0

A system partial differential equations for ''F''₂ is

:
x(1-x) \frac - xy \frac
+ _1 - (a+b_1+1) x\frac -b_1 y \frac -
a b_1 F_2(x,y) = 0

:
y(1-y) \frac - xy \frac
+ _2 - (a+b_2+1) y\frac -b_2 x \frac -
a b_2 F_2(x,y) = 0

The system have solution

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

Similarly, for ''F''₃ the following derivatives result from the definition:

:$\frac F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y)$

:$\frac F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y)$

And for ''F''₃ the following system of differential equations is obtained:

:
x(1-x) \frac + y \frac
+ - (a_1+b_1+1) x\frac -
a_1 b_1 F_3(x,y) = 0

:
y(1-y) \frac + x \frac
+ - (a_2+b_2+1) y\frac -
a_2 b_2 F_3(x,y) = 0

A system partial differential equations for ''F''₄ is

:
x(1-x) \frac - y^2 \frac
-2xy\frac + _1 - (a+b+1) x\frac - (a+b+1) y \frac -a b F_4(x,y)= 0

:
y(1-y) \frac - x^2 \frac
-2xy\frac + _2 - (a+b+1) y\frac - (a+b+1) x \frac -a b F_4(x,y)= 0

The system has solution

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

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 ''F''₁ can also be written as a one-dimensional Euler-type integral:

:$F_1(a,b_1,b_2,c; x,y) = \frac \int_0^1 t^ (1-t)^ (1-xt)^ (1-yt)^ \,\mathrmt, \quad \real \,c > \real \,a > 0 ~.$

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 ''F''₁:

:$F(\phi,k) = \int_0^\phi \frac = \sin (\phi) \,F_1(\tfrac 1 2, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad , \real \,\phi, < \frac \pi 2 ~,$

:$E(\phi, k) = \int_0^\phi \sqrt \,\mathrm \theta = \sin (\phi) \,F_1(\tfrac 1 2, \tfrac 1 2, -\tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad , \real \,\phi, < \frac \pi 2 ~,$

:$\Pi(n,k) = \int_0^ \frac = \frac \,F_1(\tfrac 1 2, 1, \tfrac 1 2, 1; n,k^2) ~.$

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 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.
* 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)
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* (see p. 224)
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* (see also ''C. R. Acad. Sci.'' 90 (1880), pp. 1119–1121 and 1267–1269)
* (there is a 2008 paperback with )

External links

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{{DEFAULTSORT:Appell Series
Hypergeometric functions
Mathematical series