# Wilson polynomials

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In mathematics, **Wilson polynomials** are a family of [orthogonal polynomials](/source/Orthogonal_polynomials) introduced by James Wilson[1] that generalize [Jacobi polynomials](/source/Jacobi_polynomials), [Hahn polynomials](/source/Hahn_polynomials), and [Charlier polynomials](/source/Charlier_polynomials).

They are defined in terms of the [generalized hypergeometric function](/source/Generalized_hypergeometric_function) and the [Pochhammer symbols](/source/Pochhammer_symbol) by

- p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n {}_4F_3\left( \begin{matrix} -n&a+b+c+d+n-1&a-t&a+t \\ a+b&a+c&a+d \end{matrix} ;1\right).

## See also

- [Askey–Wilson polynomials](/source/Askey%E2%80%93Wilson_polynomial) are a q-analogue of Wilson polynomials.

## References

1. Wilson, James A. (July 1980). ["Some Hypergeometric Orthogonal Polynomials"](http://epubs.siam.org/doi/10.1137/0511064). *SIAM Journal on Mathematical Analysis*. **11** (4): 690–701. [doi:10.1137/0511064](https://doi.org/10.1137/0511064). [ISSN 0036-1410](https://www.worldcat.org/issn/0036-1410)

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