Charlier polynomials

Short description: Orthogonal polynomials

In mathematics, Charlier polynomials, also called Poisson–Charlier polynomials, are a family of orthogonal polynomials introduced by Carl Charlier in 1905. They are given in terms of the generalized hypergeometric function by

$C_n(x;\mu )={}_2{F}_0(-n,-x;-;-1/\mu )=(-1{)}^{n}n!L_n^{(-1-x)}(-1/\mu ),$

where $L$ are generalized Laguerre polynomials. They satisfy the following orthogonality relation in the Hilbert space of square summable sequences associated with the Poisson distribution with parameter $\mu$

$e^{\mu }⟨C_n(\cdot ,\mu ),C_m(\cdot ,\mu )⟩=\sum _{x=0}^{\mathrm{\infty }}\frac{{\mu }^x}{x!}{C}_n(x;\mu )C_m(x;\mu )=e^{\mu }{\mu }^{-n}n!{\delta }_{nm},\mu >0,$

where ${\delta }_{nm}$ is the Kronecker delta. They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to Brownian motion.

Charlier polynomials have a q-analog, the q-Charlier polynomials, which are given in terms of the basic hypergeometric function by

$C_n(q^{-x};a;q)={}_2{\varphi }_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).$

See also

• Wilson polynomials, a generalization of Charlier polynomials.

References

• Charlier, C. V. L. (1905). "Über die Darstellung willkürlicher Funktionen" (in de). Arkiv för matematik, astronomi och fysik 2 (20). 

• Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. 

• Koornwinder, T. H.; Wong, R.; Koekoek, R.; Swarttouw, R. F. (2010). "Orthogonal polynomials". in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al.. NIST Handbook of Mathematical Functions. Cambridge University Press. pp. 435–484. ISBN 978-0-521-19225-5. https://dlmf.nist.gov/18.19. 

• Orthogonal Polynomials. Colloquium Publications. 23 (4th ed.). Providence, RI: American Mathematical Society. 1975. ISBN 0-8218-1023-5.