Confluent Hypergeometric Function of the First Kind
The confluent hypergeometric function of the first kind is a degenerate form of the hypergeometric
function which arises as a solution the confluent
hypergeometric differential equation. It is also known as Kummer's function of
the first kind. There are a number of other notations used for the function (Slater
1960, p. 2), including (Kummer 1836), (Airey and Webb 1918), (Humbert 1920), and (Magnus and Oberhettinger 1948). An alternate
form of the solution to the confluent
hypergeometric differential equation is known as the Whittaker
function.
where and are Pochhammer symbols.
If and are integers, , and either or , then the series yields a polynomial
with a finite number of terms. If is an integer , then is undefined. The confluent hypergeometric function
is given in terms of the Laguerre polynomial
by
(2)
(Arfken 1985, p. 755), and also has an integral representation
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