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Kummer differential equation

Kummer's equation is

x y^{''} + (b - x) y^{'} - a y = 0.

Its solution analytic at zero is the confluent hypergeometric function

M (x, a, b) = \sum_{m = 0}^{\infty} \frac{( a ) _{m}}{( b ) _{m} m !} x^{m} .

When

b

is not an integer, a second local solution is

x^{1 - b} M (x, a - b + 1, 2 - b) .

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