Wilson polynomials

In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James Wilson ^[1] that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials.

They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by

${\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\a+b&a+c&a+d\end{matrix}};1\right).}$

See also

• Askey–Wilson polynomials are a q-analogue of Wilson polynomials.

References

1. Wilson, James A. (July 1980). "Some Hypergeometric Orthogonal Polynomials". SIAM Journal on Mathematical Analysis. 11 (4): 690–701. doi:10.1137/0511064. ISSN   0036-1410.

Further reading

• Koornwinder, T.H. (2001) [1994], "Wilson polynomials", Encyclopedia of Mathematics , EMS Press