be expressed as . Relations There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples. Contiguous relations Given , the four functions are called contiguous to . The function can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of , and . This gives relations, given by identifying any two lines on the right hand side of In the notation above, , , and so on. Repeatedly applying these relations gives a linear relation between any three functions of the form (and their higher derivatives), where , are integers. There are similar relations for . Kummer's transformation Kummer's functions are also related by Kummer's transformations: . Multiplication theorem The following multiplication theorems hold true: Connection with Laguerre polynomials and similar representations In terms of Laguerre polynomials, Kummer's functions have several expansions, for example or Special cases Functions that can be expressed as special cases of the confluent hypergeometric function include: Some elementary functions where the left-hand side is not defined when is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation: (a polynomial if is a non-positive integer) for non-positive integer is a generalized Laguerre polynomial. for non-positive integer is a multiple of a generalized Laguerre polynomial, equal to when the latter exists. when is a positive integer is a closed form with powers of , equal to when the latter exists. for non-negative integer is a Bessel polynomial (see lower down). etc. Using the contiguous relation we get, for example, Bateman's function Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, in the special case the function reduces to a Bessel function: This identity is sometimes also referred to as Kummer's second transformation. Similarly When is a non-positive integer, this equals where is a Bessel polynomial. The error function can be expressed as Coulomb wave function Cunningham functions Exponential integral and related functions such as the sine integral, logarithmic integral Hermite polynomials Incomplete gamma function Laguerre polynomials Parabolic cylinder function (or Weber function) Poisson–Charlier function Toronto functions Whittaker functions are solutions of Whittaker's equation that can be expressed in terms of Kummer functions and by The general -th raw moment ( not necessarily an integer) can be expressed as In the second formula the function's