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Chapter 10:
The Confluent Hypergeometric Functions
10.1 Introduction Whereas Gauss was largely responsible for the systematic study of the hypergeometric function, E. E. Kummer (1810–1893) is the person most associated with developing properties of the related confluent hypergeometric function. Kummer published his work on this function in 1836, and since that time it has been commonly referred to as Kummer's function. Like the hypergeometric function, the confluent hypergeometric function is related to a large number of other functions. Kummer's function satisfies a second-order linear differential equation called the confluent hypergeometric equation. A second solution of this DE leads to the definition of the confluent hypergeometric function of the second kind, which is also related to many other functions. At the beginning of the twentieth century (1904), Whittaker introduced another pair of confluent hypergeometric functions that now bears his name. The Whittaker functions arise as solutions of the confluent hypergeometric equation after a transformation to Liouville's standard form of the DE.
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