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Hahn Polynomial

The orthogonal polynomials defined by

h_n^((alpha,beta))(x,N)=((-1)^n(N-x-n)_n(beta+x+1)_n)/(n!) ×_3F_2(-n,-x,alpha+N-x; N-x-n,-beta-x-n;1) =((-1)^n(N-n)_n(beta+1)_n)/(n!) ×_3F_2(-n,-x,alpha+beta+n+1; beta+1,1-N;1),
(1)

where (x)_n is the Pochhammer symbol and _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function (Koepf 1998). The first few are given by

h_0^((alpha,beta))(x,N)=1
(2)
h_1^((alpha,beta))(x,N)=x(alpha+beta+2)-(N-1)(beta+1).
(3)

Koekoek and Swarttouw (1998) define another Hahn polynomial

Q_n(x;alpha,beta,N)=_3F_2(-n,n+alpha+beta+1,-x; alpha+1,-N;1),
(4)

the dual Hahn polynomial

R_n(lambda(x);gamma,delta,N)=_3F_2(-n,-x,x+gamma+delta+1; gamma+1,-N;1),
(5)

the continuous Hahn polynomial

p_n(x;a,b,c,d)=i^n((a+c)_n(a+d)_n)/(n!) ×_3F_2(-n,n+a+b+c+d-1,a+ix; a+c,a+d;1),
(6)

and the continuous dual Hahn polynomial

(S_n(x^2;a,b,c))/((a+b)_n(a+c)_n)=_3F_2(-n,a+ix,a-ix; a+b,a+c;1),
(7)

for n=0, 1, ..., N, and where

lambda(x)=x(x+gamma+delta+1).
(8)

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References

Koekoek, R. and Swarttouw, R. F. "Continuous Dual Hahn," "Continuous Hahn," "Hahn," and "Dual Hahn." §1.3-1.6 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. 29-36, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998.

Referenced on Wolfram|Alpha

Hahn Polynomial

Cite this as:

Weisstein, Eric W. "Hahn Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HahnPolynomial.html

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