mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 class ...

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, 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 abo ...

\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 ...

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 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 ...

References

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

See also

References