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Special Functions by Roderick Wong, Richard Beals

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6

Confluent hypergeometric functions

The confluent hypergeometric equation

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has one solution, the Kummer function M(a, c; x), with value 1 at the origin, and a second solution, x1−c M(a + 1 − c, 2 − c; x), which is ∼ x1−c at the origin, provided that c is not an integer. A particular linear combination of the two gives a solution U (a, c; x) ∼ xa as x → +∞. The Laguerre polynomials are particular cases, corresponding to particular values of the parameters. Like the Laguerre polynomials, the general solutions satisfy a number of linear relations involving derivatives and different values of the parameters a and c. Special consideration is required ...

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