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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:

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

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

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

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:

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

the function F3 for |x| < 1, |y| < 1 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|½ + |y|½ < 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:

$$\frac {\partial} {\partial x} F_1(a,b_1,b_2,c; x,y) = \frac {a b_1} {c} F_1(a+1,b_1+1,b_2,c+1; x,y) ~,$$

$$\frac {\partial} {\partial y} F_1(a,b_1,b_2,c; x,y) = \frac {a b_2} {c} F_1(a+1,b_1,b_2+1,c+1; x,y) ~.$$

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

$$\left( x(1-x) \frac {\partial^2} {\partial x^2} + y(1-x) \frac {\partial^2} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial} {\partial x} - b_1 y \frac {\partial} {\partial y} - a b_1 \right) F_1(x,y) = 0 ~,$$

$$\left( y(1-y) \frac {\partial^2} {\partial y^2} + x(1-y) \frac {\partial^2} {\partial x \partial y} + [c - (a+b_2+1) y] \frac {\partial} {\partial y} - b_2 x \frac {\partial} {\partial x} - a b_2 \right) F_1(x,y) = 0 ~.$$

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:

$$\left( x(1-x) \frac {\partial^2} {\partial x^2} + y \frac {\partial^2} {\partial x \partial y} + [c - (a_1+b_1+1) x] \frac {\partial} {\partial x} - a_1 b_1 \right) F_3(x,y) = 0 ~,$$

$$\left( y(1-y) \frac {\partial^2} {\partial y^2} + x \frac {\partial^2} {\partial x \partial y} + [c - (a_2+b_2+1) y] \frac {\partial} {\partial y} - a_2 b_2 \right) F_3(x,y) = 0 ~.$$

Integral representations

The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only. However, Émile Picard (1880) 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
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.

References

Appell, Paul (1880). "Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles" (in French). Comptes rendus hebdomadaires des séances de l'Académie des sciences 90: 296–298 and 731–735.
Appell, Paul (1882). "Sur les fonctions hypergéométriques de deux variables" (in French). Journal de Mathématiques Pures et Appliquées. (3ème série) 8: 173–216.
Appell, Paul; Kampé de Fériet, Joseph (1926) (in French). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars. (see p. 14)
Askey, R. A.; Daalhuis, Adri B. Olde (2010), "Appell series", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248
Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I. New York: McGraw–Hill. (see p. 224)
Gradshteyn, Izrail' Solomonovich; Ryzhik, Iosif Moiseevich (1971) (in Russian). Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] (5th ed.). Moscow: Nauka. (see Chapter 9.18)
Humbert, Pierre (1920). "Sur les fonctions hypercylindriques" (in French). Comptes rendus hebdomadaires des séances de l'Académie des sciences 171: 490–492.
Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili" (in Italian). Rendiconti del Circolo Matematico di Palermo 7: 111–158. doi:10.1007/BF03012437.
Picard, Émile (1880). "Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques" (in French). Comptes rendus hebdomadaires des séances de l'Académie des sciences 90: 1267–1269.
Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)