mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

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. fro ...

. 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, which ...

by :

C_n(x; \mu)= _2F_0(-n,-x;-;-1/\mu)=(-1)^n n! L_n^\left(-\frac 1 \mu \right),

where

L

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 abou ...

\sum_^\infty \frac C_n(x; \mu)C_m(x; \mu)=\mu^ e^\mu n! \delta_, \quad \mu>0.

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 n ...

related to the

Poisson process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...

, 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 a ...

relate to the

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

References

* C. V. L. Charlier (1905–1906) ''Über die Darstellung willkürlicher Funktionen'', Ark. Mat. Astr. och Fysic 2, 20. * * Orthogonal polynomials {{polynomial-stub

See also

References