Confluent hypergeometric function

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In mathematics, there are two types of functions known as confluent hypergeometric functions. One is the family of solutions to a differential equation known as Kummer's equation, and is known as "Kummer's confluent hypergeometric function", or simply Kummer's function (for Ernst Kummer). The other is the family of solutions to a differential equation known as Whittaker's equation, and is known as Whittakers's function (for E. T. Whittaker). Note also that there is a different but unrelated Kummer's function bearing the same name.

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.

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[edit] Kummer's equation

Kummer's equation is

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

It has two linearly independent solutions M(a,b,z) and U(a,b,z).

Kummer's function (first kind) is given by

M(a,b,z)={\,}_1F_1(a;b;z)= \sum_{n=0}^\infty \frac {(a)_n z^n} {(b)_n n!}\,\!.

where {\,}_1F_1(a;b;z) is a hypergeometric series and (a)n = a(a + 1)(a + 2)...(a + n − 1) is the rising factorial.

The other solution is Kummer's function (second kind):

U(a,b,z)=\frac{\pi}{\sin\pi b} \left[ \frac{M(a,b,z)} {\Gamma(1+a-b)\Gamma(b)} - z^{1-b} \frac{M(1+a-b, 2-b,z)}{\Gamma(a) \Gamma(2-b)} \right].

[edit] Kummer's function (1st kind)

The derivative of Kummer's function M  is given by:

\frac{d}{dz}\,M(a,b,z) = \frac{a}{b}\,M(a+1,b+1,z)

From which follows, by induction, that:

\frac{d^n}{dz^n}M(a,b,z)=\frac{(a)_n}{(b)_n}M(a+n,b+n,z)

Kummer's functions are also related by Kummer's transformations:

M(a,b,z) = e^z\,M(b-a,b,-z)

[edit] Kummer's function (2nd kind)

The derivative of Kummer's function U  is given by:

\frac{d}{dz}\,U(a,b,z) = -a\,U(a+1,b+1,z)

[edit] References

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