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Poisson-Charlier Polynomial

The Poisson-Charlier polynomials c_k(x;a) form a Sheffer sequence with

g(t)=e^(a(e^t-1))
(1)
f(t)=a(e^t-1),
(2)

giving the generating function

sum_(k=0)^infty(c_k(x;a))/(k!)t^k=e^(-t)((a+t)/a)^x.
(3)

The Sheffer identity is

c_n(x+y;a)=sum_(k=0)^n(n; k)a^(k-n)c_k(y;a)(x)_(n-k),
(4)

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

c_(n+1)(x;a)=a^(-1)xc_n(x-1;a)-c_n(x;a).
(5)

These polynomials belong to the distribution dalpha(x) where alpha(x) is a step function with jump

j(x)=e^(-a)a^x(x!)^(-1)
(6)

at x=0, 1, ... for a>0. They are given by the formulas

c_n(x;a)=sum_(nu=0)^(n)(-1)^(n-nu)(n; nu)nu!a^(-nu)(x; nu)
(7)
=sum_(k=0)^(n)(n; k)(-1)^(n-k)a^(-k)(x)_k
(8)
=a^n(-1)^n[j(x)]^(-1)Delta^nj(x-n)
(9)
=a^(-n)n!L_n^(x-n)(a)
(10)
=sum_(j=0)^(n)x^jsum_(k=0)^(n)(n; k)(-1)^(n-k)a^(-k)s(k,j)
(11)

where (n; k) is a binomial coefficient, (x)_n is a falling factorial, L_n^k(x) is an associated Laguerre polynomial, s(n,m) is a Stirling number of the first kind, and

Deltaf(x)=f(x+1)-f(x)
(12)
Delta^nf(x)=Delta[Delta^(n-1)f(x)]=f(x+n)-(n; 1)f(x+n-1)+...+(-1)^nf(x).
(13)

They are normalized so that

sum_(k=0)^inftyj(k)c_n(k;a)c_m(k;a)=a^(-n)n!delta_(nm),
(14)

where delta_(nm) is the delta function.

The first few polynomials are

c_0(x;a)=1
(15)
c_1(x;a)=-(a-x)/a
(16)
c_2(x;a)=(a^2-x-2ax+x^2)/(a^2)
(17)
c_3(x;a)=-(a^3-2x-3ax-3a^2x+3x^2+3ax^2-x^3)/(a^3).
(18)

See also

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.

Referenced on Wolfram|Alpha

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

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