Humbert series

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 Pierre Humbert (1920).

Definitions

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

[math]\displaystyle{ \Phi_1(a,b,c;x,y) = F_1(a,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, }[/math]

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

[math]\displaystyle{ (q)_n = q\,(q+1) \cdots (q+n-1) = \frac{\Gamma(q+n)}{\Gamma(q)}~, }[/math]

where the second equality is true for all complex [math]\displaystyle{ q }[/math] except [math]\displaystyle{ q=0,-1,-2,\ldots }[/math].

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:

[math]\displaystyle{ \Phi_1(a,b,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} e^{yt} \,\mathrm{d}t, \quad \real \,c \gt \real \,a \gt 0 ~. }[/math]

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:

[math]\displaystyle{ \Phi_2(b_1,b_2,c;x,y) = F_1(-,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, }[/math]

the function Φ₃ for all x, y by the series:

[math]\displaystyle{ \Phi_3(b,c;x,y) = \Phi_2(b,-,c;x,y) = F_1(-,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, }[/math]

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

[math]\displaystyle{ \Psi_1(a,b,c_1,c_2;x,y) = F_2(a,b,-,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~, }[/math]

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

[math]\displaystyle{ \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_{m,n=0}^\infty \frac{(a)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~, }[/math]

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

[math]\displaystyle{ \Xi_1(a_1,a_2,b,c;x,y) = F_3(a_1,a_2,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, }[/math]

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

[math]\displaystyle{ \Xi_2(a,b,c;x,y) = \Xi_1(a,-,b,c;x,y) = F_3(a,-,b,-,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_m (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~. }[/math]

Related 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

• Appell, Paul; Kampé de Fériet, Joseph (1926) (in French). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite. Paris: Gauthier–Villars.  (see p. 126)
• Bateman, H.; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I. New York: McGraw–Hill. http://apps.nrbook.com/bateman/Vol1.pdf.  (see p. 225)
• "9.26." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. ISBN 978-0-12-384933-5. 
• Humbert, Pierre (1920). "Sur les fonctions hypercylindriques" (in French). Comptes rendus hebdomadaires des séances de l'Académie des sciences 171: 490–492.