In
mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the cl ...
introduced by
Carl Charlier
Carl Vilhelm Ludwig Charlier (1 April 1862 – 4 November 1934) was a Swedish astronomer. His parents were Emmerich Emanuel and Aurora Kristina (née Hollstein) Charlier.
Career
Charlier was born in Östersund. He received his Ph.D. from ...
.
They are given in terms of the
generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ...
by
:
where
are
generalized Laguerre polynomials. They satisfy the
orthogonality relation
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...
:
They form a
Sheffer sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are na ...
related to the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
, similar to how
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well ...
relate to the
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
See also
*
Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by
that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials.
They are defined in terms of the generalized hypergeometric function and the ...
, a generalization of Charlier polynomials.
References
* C. V. L. Charlier (1905–1906) ''Über die Darstellung willkürlicher Funktionen'', Ark. Mat. Astr. och Fysic 2, 20.
*
*
Orthogonal polynomials
{{Algebra-stub