Humbert series explained

In mathematics, Humbert series are a set of seven hypergeometric series Φ₁, Φ₂, Φ₃, Ψ₁, Ψ₂, Ξ₁, Ξ₂ of two variables that generalize Kummer's confluent hypergeometric series ₁F₁ of one variable and the confluent hypergeometric limit function ₀F₁ of one variable. The first of these double series was introduced by .

Definitions

The Humbert series Φ₁ is defined for |x| < 1 by the double series:

\Phi _{1(a,b,c;x,y)} = F _{1(a,b,-,c;x,y)} = \sum ^{inf ty} _{m,n =0} ((a) _{m+n} (b) _{m})/((c) _{m+n} m! n!) x ^{m} y ^{n} ~,

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

(q) _{n} = q(q +1 ) … (q + n -1 ) = (\Gamma(q + n))/(\Gamma(q)) ~,

where the second equality is true for all complex

q except q =0 , -1 , -2 ,\ldots . For other values of x the function Φ₁ can be defined by analytic continuation.

The Humbert series Φ₁ can also be written as a one-dimensional Euler-type integral:

\Phi _{1(a,b,c;x,y)} = (\Gamma(c))/(\Gamma(a) \Gamma(c - a)) \int ^{1} _{0} t ^{a-1} ( 1- t) ^{c-a-1} ( 1- xt) ^{-b} e ^{yt} d t, \real c > \real a > 0 ~.

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

Similarly, the function Φ₂ is defined for all x, y by the series:

\Phi _{2(b} _{1,b} _{2,c;x,y)} = F _{1(-,b} _{1,b} _{2,c;x,y)} = \sum ^{inf ty} _{m,n =0} ((b _{1)} _{m} (b _{2)} _{n})/((c) _{m+n} m! n!) x ^{m} y ^{n} ~, the function Φ₃ for all x, y by the series:\Phi _{3(b,c;x,y)} = \Phi _{2(b,-,c;x,y)} = F _{1(-,b,-,c;x,y)} = \sum ^{inf ty} _{m,n =0} ((b) _{m})/((c) _{m+n} m! n!) x ^{m} y ^{n} ~,

the function Ψ₁ for |x| < 1 by the series:

\Psi _{1(a,b,c} _{1,c} _{2;x,y)} = F _{2(a,b,-,c} _{1,c} _{2;x,y)} = \sum ^{inf ty} _{m,n =0} ((a) _{m+n} (b) _{m})/((c _{1)} _{m} (c _{2)} _{n} m! n!) x ^{m} y ^{n} ~,

the function Ψ₂ for all x, y by the series:

\Psi _{2(a,c} _{1,c} _{2;x,y)} = \Psi _{1(a,-,c} _{1,c} _{2;x,y)} = F _{2(a,-,-,c} _{1,c} _{2;x,y)} = F _{4(a,-,c} _{1,c} _{2;x,y)} = \sum ^{inf ty} _{m,n =0} ((a) _{m+n})/((c _{1)} _{m} (c _{2)} _{n} m! n!) x ^{m} y ^{n} ~,

the function Ξ₁ for |x| < 1 by the series:

\Xi _{1(a} _{1,a} _{2,b,c;x,y)} = F _{3(a} _{1,a} _{2,b,-,c;x,y)} = \sum ^{inf ty} _{m,n =0} ((a _{1)} _{m} (a _{2)} _{n} (b) _{m})/((c) _{m+n} m! n!) x ^{m} y ^{n} ~,

and the function Ξ₂ for |x| < 1 by the series:

\Xi _{2(a,b,c;x,y)} = \Xi _{1(a,-,b,c;x,y)} = F _{3(a,-,b,-,c;x,y)} = \sum ^{inf ty} _{m,n =0} ((a) _{m} (b) _{m})/((c) _{m+n} m! n!) x ^{m} y ^{n} ~.

Related series

See main article: Appell series.

There are four related series of two variables, F₁, F₂, F₃, and F₄, which generalize Gauss's hypergeometric series ₂F₁ of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.

References

• Book: Appell . Paul . Paul Émile Appell . Kampé de Fériet . Joseph . Joseph Kampé de Fériet . Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite . French . Paris . Gauthier–Villars . 1926 . 52.0361.13 . (see p. 126)
• Book: H. . Bateman . Harry Bateman . A. . Erdélyi . Arthur Erdélyi . Higher Transcendental Functions, Vol. I . New York . McGraw–Hill . 1953 . (see p. 225)
• Book: Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . Victor Hugo . Moll . Victor Hugo Moll . Scripta Technica, Inc. . Table of Integrals, Series, and Products . . 2015 . October 2014 . 8 . English . 978-0-12-384933-5 . 2014010276 . 2016-02-21-->. Gradshteyn and Ryzhik . 9.26. .
• Humbert . Pierre . Pierre Humbert (mathematician) . Sur les fonctions hypercylindriques . French . Comptes rendus hebdomadaires des séances de l'Académie des sciences . 1920 . 171 . 490 - 492 . 47.0348.01 .