Poisson-Charlier Polynomial
المؤلف:
Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G.
المصدر:
Higher Transcendental Functions, Vol. 2. New York: Krieger
الجزء والصفحة:
...
22-9-2019
1951
Poisson-Charlier Polynomial
The Poisson-Charlier polynomials 
form a Sheffer sequence with
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
where 
is a binomial coefficient, 
is a falling factorial, 
is an associated Laguerre polynomial, 
is a Stirling number of the first kind, and
They are normalized so that
 |
(14)
|
where 
is the delta function.
The first few polynomials are
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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