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Special Functions by Roderick Wong, Richard Beals

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3

Second-order differential equations

As noted in Chapter 1, most of the functions commonly known as “special functions” are solutions of second-order linear differential equations. These equations occur naturally in certain physical and mathematical contexts. In a certain sense there are exactly two (families of) equations in question: the confluent hypergeometric equation (Kummer’s equation)

image

(3.0.1)

with indices (a, c), and the hypergeometric equation

image

(3.0.2)

with indices (a, b, c), where a, b, c are constants. The various ...

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