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)

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.