Polynomials
which form the Sheffer
sequence for
and have generating function
![sum_(k=0)^infty(m_k(x;b,c))/(k!)t^k=(1-t/c)^x(1-t)^(-x-b).](/images/equations/MeixnerPolynomialoftheFirstKind/NumberedEquation1.svg) |
(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)),](/images/equations/MeixnerPolynomialoftheFirstKind/NumberedEquation2.svg) |
(4)
|
where
is the Pochhammer symbol (Koepf 1998, p. 115).
The first few are
![m_2(x;b,c)
=(b(b+1)c^2+(c-1)(2bc+c+1)x+(c-1)^2x^2)/(c^2).](/images/equations/MeixnerPolynomialoftheFirstKind/NumberedEquation3.svg) |
(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).](/images/equations/MeixnerPolynomialoftheFirstKind/NumberedEquation4.svg) |
(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
-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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