In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the

confluent hypergeometric equation Confluent, Inc. is an American technology company headquartered in Mountain View, California. Confluent was founded by Jay Kreps, Jun Rao and Neha Narkhede on September 23, 2014, in order to commercialize an open-source streaming platform Apa ...

introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation that has a finite kernel and is a ...

s over

local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

s, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL₂(R). Whittaker's equation is :

\frac+\left(-\frac+\frac+\frac\right)w=0.

It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions ''M''_κ,μ(''z''), ''W''_κ,μ(''z''), defined in terms of Kummer's

confluent hypergeometric functions In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular s ...

''M'' and ''U'' by :

M_\left(z\right) = \exp\left(-z/2\right)z^M\left(\mu-\kappa+\tfrac, 1+2\mu, z\right)

W_\left(z\right) = \exp\left(-z/2\right)z^U\left(\mu-\kappa+\tfrac, 1+2\mu, z\right).

The Whittaker function

W_(z)

is the same as those with opposite values of , in other words considered as a function of at fixed and it is

even function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

s. When and are real, the functions give real values for real and imaginary values of . These functions of play a role in so-called

Kummer space Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chris ...

s. Sections 55-57. Whittaker functions appear as coefficients of certain representations of the group SL₂(R), called

Whittaker model In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named afte ...

References

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Further reading