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Meixner Polynomial of the First Kind

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Polynomials m_k(x;beta,c) which form the Sheffer sequence for

g(t)=((1-c)/(1-ce^t))^beta
(1)
f(t)=(1-e^t)/(c^(-1)-e^t)
(2)

and have generating function

sum_(k=0)^infty(m_k(x;b,c))/(k!)t^k=(1-t/c)^x(1-t)^(-x-b).
(3)

The are given in terms of the hypergeometric series by

m_n(x;gamma,mu)=(gamma)_n_2F_1(-n,-x;gamma;1-mu^(-1)),
(4)

where (x)_n is the Pochhammer symbol (Koepf 1998, p. 115). The first few are

m_0(x;b,c)=1
(5)
m_1(x;b,c)=b+x(1-1/c)
(6)
m_2(x;b,c) =(b(b+1)c^2+(c-1)(2bc+c+1)x+(c-1)^2x^2)/(c^2).
(7)

Koekoek and Swarttouw (1998) defined the Meixner polynomials without the Pochhammer symbol as

M_n^'(x;beta,c)=_2F_1(-n,-x;beta;1-1/c).
(8)

The Krawtchouk polynomials are a special case of the Meixner polynomials of the first kind.

See also

Krawtchouk Polynomial, Meixner Polynomial of the Second Kind, Sheffer Sequence

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References

Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, p. 175, 1978.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, pp. 224-225, 1981.Koekoek, R. and Swarttouw, R. F. "Meixner." §1.9 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 45-46, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 35, 1975.

Referenced on Wolfram|Alpha

Meixner Polynomial of the First Kind

Cite this as:

Weisstein, Eric W. "Meixner Polynomial of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MeixnerPolynomialoftheFirstKind.html

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