confluent hypergeometric function

In mathematics, the confluent hypergeometric function is formed from hypergeometric series. It occurs in two forms, as Kummer's function (for Ernst Kummer) and as Whittaker's function (for E. T. Whittaker). Note also that there is a different Kummer's function bearing the same name but unrelated to this. Kummer's equation is

z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0.

It has two independent solutions

M(a,b,z)

and

U(a,b,z)

. Kummer' function is given by

M(a,b,z)=\sum_{n=0}^\infty \frac {(a)_n z^n} {(b)_n n!}

where

(a)_n=a(a+1)(a+2)...(a+n-1)

is the rising factorial. and

U(a,b,z)=\frac{\pi}{\sin\pi b} \left {\Gamma(1+a-b)\Gamma(b)} - z^{1-b}

\frac{M(1+a-b, 2-b,z)}{\Gamma(a) \Gamma(2-b)} \right. The term confluent refers to the singular points of the differential equation, on the Riemann sphere. Where the usual hypergeometric equation has three singular points (in general position), confluence implies cases of degeneration by singularities being brought together by a limiting process.

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 See chapter 13.

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