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Poisson-Charlier Polynomial

The Poisson-Charlier polynomials form a Sheffer sequence with

 (1) (2)

giving the generating function

 (3)

The Sheffer identity is

 (4)

where is a falling factorial (Roman 1984, p. 121). The polynomials satisfy the recurrence relation

 (5)

These polynomials belong to the distribution where is a step function with jump

 (6)

at , 1, ... for . They are given by the formulas

 (7) (8) (9) (10) (11)

where is a binomial coefficient, is a falling factorial, is an associated Laguerre polynomial, is a Stirling number of the first kind, and

 (12) (13)

They are normalized so that

 (14)

where is the delta function.

The first few polynomials are

 (15) (16) (17) (18)

Laguerre Polynomial, Poisson-Charlier Function, Sheffer Sequence

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, p. 226, 1981.Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, p. 473, 1965.Roman, S. "The Poisson-Charlier Polynomials." §4.3.3 in The Umbral Calculus. New York: Academic Press, pp. 119-122, 1984.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 34-35, 1975.

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Poisson-Charlier Polynomial

Cite this as:

Weisstein, Eric W. "Poisson-Charlier Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Poisson-CharlierPolynomial.html