In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by
![{\displaystyle p_{n}(x;a,b,c,d)=i^{n}{\frac {(a+c)_{n}(a+d)_{n}}{n!}}{}_{3}F_{2}\left({\begin{array}{c}-n,n+a+b+c+d-1,a+ix\\a+c,a+d\end{array}};1\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b802c046e321b7fdf98f4ac26a5b55cd0d65b299)
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the Hahn polynomials Qn(x;a,b,c), and the continuous dual Hahn polynomials Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials Qn(x;α,β, N;q), and so on.
Orthogonality
The continuous Hahn polynomials pn(x;a,b,c,d) are orthogonal with respect to the weight function
![{\displaystyle w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b79c3d28ec72f171748e534f8ec33142a1ea08)
In particular, they satisfy the orthogonality relation[1][2][3]
![{\displaystyle {\begin{aligned}&{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{m}(x;a,b,c,d)\,p_{n}(x;a,b,c,d)\,dx\\&\qquad \qquad ={\frac {\Gamma (n+a+c)\,\Gamma (n+a+d)\,\Gamma (n+b+c)\,\Gamma (n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma (n+a+b+c+d-1)}}\,\delta _{nm}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b279dbe686c8a7391028a6137bf6d5a6dce251db)
for
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Recurrence and difference relations
The sequence of continuous Hahn polynomials satisfies the recurrence relation[4]
![{\displaystyle xp_{n}(x)=p_{n+1}(x)+i(A_{n}+C_{n})p_{n}(x)-A_{n-1}C_{n}p_{n-1}(x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/774feb64bbb83a12b2cdf286f91027a0e6408856)
![{\displaystyle {\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d75c9411446a435cb73af362c29331995248b1bd)
The continuous Hahn polynomials are given by the Rodrigues-like formula[5]
![{\displaystyle {\begin{aligned}&\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{n}(x;a,b,c,d)\\&\qquad ={\frac {(-1)^{n}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(\Gamma \left(a+{\frac {n}{2}}+ix\right)\,\Gamma \left(b+{\frac {n}{2}}+ix\right)\,\Gamma \left(c+{\frac {n}{2}}-ix\right)\,\Gamma \left(d+{\frac {n}{2}}-ix\right)\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/421bd8b88a1cdc31210ae4271c72e709a9d292e4)
Generating functions
The continuous Hahn polynomials have the following generating function:[6]
![{\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }{\frac {\Gamma (n+a+b+c+d)\,\Gamma (a+c+1)\,\Gamma (a+d+1)}{\Gamma (a+b+c+d)\,\Gamma (n+a+c+1)\,\Gamma (n+a+d+1)}}(-it)^{n}p_{n}(x;a,b,c,d)\\&\qquad =(1-t)^{1-a-b-c-d}{}_{3}F_{2}\left({\begin{array}{c}{\frac {1}{2}}(a+b+c+d-1),{\frac {1}{2}}(a+b+c+d),a+ix\\a+c,a+d\end{array}};-{\frac {4t}{(1-t)^{2}}}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7322e4de5bc87d055e20e6908fb98bb7ed70831f)
A second, distinct generating function is given by
![{\displaystyle \sum _{n=0}^{\infty }{\frac {\Gamma (a+c+1)\,\Gamma (b+d+1)}{\Gamma (n+a+c+1)\,\Gamma (n+b+d+1)}}t^{n}p_{n}(x;a,b,c,d)=\,_{1}F_{1}\left({\begin{array}{c}a+ix\\a+c\end{array}};-it\right)\,_{1}F_{1}\left({\begin{array}{c}d-ix\\b+d\end{array}};it\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7965765834b1c0a74de3c272411bbbf1ad476dc)
Relation to other polynomials
- The Wilson polynomials are a generalization of the continuous Hahn polynomials.
- The Bateman polynomials Fn(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by
![{\displaystyle p_{n}\left(x;{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}}\right)=i^{n}n!F_{n}\left(2ix\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/106ab17dd8548c82fa6be4483f313559d417d29f)
- The Jacobi polynomials Pn(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:[7]
![{\displaystyle P_{n}^{(\alpha ,\beta )}=\lim _{t\to \infty }t^{-n}p_{n}\left({\tfrac {1}{2}}xt;{\tfrac {1}{2}}(\alpha +1-it),{\tfrac {1}{2}}(\beta +1+it),{\tfrac {1}{2}}(\alpha +1+it),{\tfrac {1}{2}}(\beta +1-it)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3b3763e374933b13bbb58c2120091629b32b99)
References
- ^ Koekoek, Lesky, & Swarttouw (2010), p. 200.
- ^ Askey, R. (1985), "Continuous Hahn polynomials", J. Phys. A: Math. Gen. 18: pp. L1017-L1019.
- ^ Andrews, Askey, & Roy (1999), p. 333.
- ^ Koekoek, Lesky, & Swarttouw (2010), p. 201.
- ^ Koekoek, Lesky, & Swarttouw (2010), p. 202.
- ^ Koekoek, Lesky, & Swarttouw (2010), p. 202.
- ^ Koekoek, Lesky, & Swarttouw (2010), p. 203.
- Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 2: 4–34, doi:10.1002/mana.19490020103, 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, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge: Cambridge University Press, ISBN 978-0-521-62321-6