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 ₁''F''₁ of one variable and the confluent hypergeometric limit function ₀''F''₁ 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:

:$\Phi_1(a,b,c;x,y) = F_1(a,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

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

:$(q)_n = q\,(q+1) \cdots (q+n-1) = \frac~,$

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:

:$\Phi_1(a,b,c;x,y) = \frac \int_0^1 t^ (1-t)^ (1-xt)^ e^ \,\mathrmt, \quad \real \,c > \real \,a > 0 ~.$

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:

:$\Phi_2(b_1,b_2,c;x,y) = F_1(-,b_1,b_2,c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

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

:$\Phi_3(b,c;x,y) = \Phi_2(b,-,c;x,y) = F_1(-,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

the function Ψ₁ for , ''x'', < 1 by the series:

:$\Psi_1(a,b,c_1,c_2;x,y) = F_2(a,b,-,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

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

:$\Psi_2(a,c_1,c_2;x,y) = \Psi_1(a,-,c_1,c_2;x,y) = F_2(a,-,-,c_1,c_2;x,y) = F_4(a,-,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

the function Ξ₁ for , ''x'', < 1 by the series:

:$\Xi_1(a_1,a_2,b,c;x,y) = F_3(a_1,a_2,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$

and the function Ξ₂ for , ''x'', < 1 by the series:

:$\Xi_2(a,b,c;x,y) = \Xi_1(a,-,b,c;x,y) = F_3(a,-,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~.$

Related series

*
:There are four related series of two variables, ''F''₁, ''F''₂, ''F''₃, and ''F''₄, which generalize Gauss's hypergeometric series ₂''F''₁ of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.

References

* (see p. 126)
* (see p. 225)
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{{DEFAULTSORT:Humbert Series
Hypergeometric functions
Mathematical series