# Hahn polynomials

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In mathematics, the **Hahn polynomials** are a family of [orthogonal polynomials](/source/Orthogonal_polynomials) in the [Askey scheme](/source/Askey_scheme) of hypergeometric orthogonal polynomials, introduced by [Pafnuty Chebyshev](/source/Pafnuty_Chebyshev) in 1875 (Chebyshev 1907) and rediscovered by [Wolfgang Hahn](/source/Wolfgang_Hahn) (Hahn 1949). The **Hahn class** is a name for special cases of Hahn polynomials, including Hahn polynomials, [Meixner polynomials](/source/Meixner_polynomials), [Krawtchouk polynomials](/source/Krawtchouk_polynomials), and [Charlier polynomials](/source/Charlier_polynomials). Sometimes the Hahn class is taken to include [limiting cases](/source/Limiting_case_(mathematics)) of these polynomials, in which case it also includes the [classical orthogonal polynomials](/source/Classical_orthogonal_polynomials).

Hahn polynomials are defined in terms of [generalized hypergeometric functions](/source/Generalized_hypergeometric_function) by

- Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\

txt, 14 give a detailed list of their properties.

If \alpha = \beta = 0, these polynomials are identical to the [discrete Chebyshev polynomials](/source/Discrete_Chebyshev_polynomials) except for a scale factor.

Closely related polynomials include the [dual Hahn polynomials](/source/Dual_Hahn_polynomials) *R**n*(*x*;γ,δ,*N*), the [continuous Hahn polynomials](/source/Continuous_Hahn_polynomials) *p**n*(*x*,*a*,*b*, *a*, *b*), and the [continuous dual Hahn polynomials](/source/Continuous_dual_Hahn_polynomials) *S**n*(*x*;*a*,*b*,*c*). These polynomials all have *q*-analogs with an extra parameter *q*, such as the [q-Hahn polynomials](/source/Q-Hahn_polynomials) *Q**n*(*x*;α,β, *N*;*q*), and so on.

## Orthogonality

- \sum_{x=0}^{N-1} Q_n(x)Q_m(x)\rho(x)=\frac{1}{\pi_n}\delta_{m,n},

- \sum_{n=0}^{N-1}Q_n(x)Q_n(y)\pi_n=\frac{1}{\rho(x)}\delta_{x,y}

where *δx,y* is the Kronecker delta function and the weight functions are

- \rho(x)=\rho(x;\alpha;\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-1-x}{N-1-x}/\binom{N+\alpha+\beta}{N-1}

and

- \pi_n=\pi_n(\alpha,\beta,N)=\binom{N-1}{n}\frac{2n+\alpha+\beta+1}{\alpha+\beta+1} \frac{\Gamma(\beta+1,n+\alpha+1,n+\alpha+\beta+1)}{\Gamma(\alpha+1,\alpha+\beta+1,n+\beta+1,n+1)}/\binom{N+\alpha+\beta+n}{n}.

## Relation to other polynomials

- [Racah polynomials](/source/Racah_polynomials) are a generalization of Hahn polynomials

## References

- Chebyshev, P. (1907), ["Sur l'interpolation des valeurs équidistantes"](https://archive.org/stream/uvresdepltcheby01chebgoog#page/n250), "Oeuvres de P. L. Tchebychef", Vol. 2, pp. 219–242, Reprinted by Chelsea
- Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", *[Mathematische Nachrichten](/source/Mathematische_Nachrichten)*. **2**: 4–34, [doi:10.1002/mana.19490020103](https://doi.org/10.1002/mana.19490020103). [ISSN 0025-584X](https://www.worldcat.org/issn/0025-584X). MR 0030647
- Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), *Hypergeometric orthogonal polynomials and their q-analogues*, Springer Monographs in Mathematics, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi:10.1007/978-3-642-05014-5](https://doi.org/10.1007/978-3-642-05014-5). ISBN 978-3-642-05013-8. MR 2656096

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