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Humbert series

In mathematics, Humbert series are a set of seven hypergeometric series Φ<sub>1</sub>, Φ<sub>2</sub>, Φ<sub>3</sub>, Ψ<sub>1</sub>, Ψ<sub>2</sub>, Ξ<sub>1</sub>, Ξ<sub>2</sub> of two variables that generalize Kummer's confluent hypergeometric series <sub>1</sub>F<sub>1</sub> of one variable and the confluent hypergeometric limit function <sub>0</sub>F<sub>1</sub> of one variable. The first of these double series was introduced by .

Definitions

The Humbert series Φ<sub>1</sub> is defined for |x| < 1 by the double series:

where the Pochhammer symbol (q)<sub>n</sub> represents the rising factorial:

where the second equality is true for all complex except .

For other values of x the function Φ<sub>1</sub> can be defined by analytic continuation.

The Humbert series Φ<sub>1</sub> can also be written as a one-dimensional Euler-type integral:

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

Similarly, the function Φ<sub>2</sub> is defined for all x, y by the series:

the function Φ<sub>3</sub> for all x, y by the series:

the function Ψ<sub>1</sub> for |x| < 1 by the series:

the function Ψ<sub>2</sub> for all x, y by the series:

the function Ξ<sub>1</sub> for |x| < 1 by the series:

and the function Ξ<sub>2</sub> for |x| < 1 by the series:

Related series

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

References

  • (see p.&nbsp;126)
  • (see p.&nbsp;225)