Confluent hypergeometric functions solve the Kummer differential equation . The regular solution is the confluent hypergeometric function of the first kind; a second standard solution is conventionally denoted . The family arises by coalescing singular points of the Gauss hypergeometric equation.
For not a nonpositive integer, the regular solution normalized by has the power seriesHere is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.
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The Confluent hypergeometric function is a special function that arises in various areas of mathematics and physics, particularly in the context of solving differential equations. It is a limit case of the more general hypergeometric function and is particularly useful in situations where the parameters of the hypergeometric function simplify, leading to the confluent form.