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
p
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n
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3
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{\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)}
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 R n x ;γ,δ,N ), the Hahn polynomials Q n x ;a ,b ,c ), and the 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 Q n x ;α,β, N ;q ), and so on.
The continuous Hahn polynomials p n x ;a ,b ,c ,d ) are orthogonal with respect to the weight function
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{\displaystyle w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix).}
In particular, they satisfy the orthogonality relation[ 1] [ 2] [ 3]
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2
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∞
∞
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{\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}}}
for
ℜ
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{\displaystyle \Re (a)>0}
ℜ
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{\displaystyle \Re (b)>0}
ℜ
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{\displaystyle \Re (c)>0}
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{\displaystyle \Re (d)>0}
c
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{\displaystyle c={\overline {a}}}
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{\displaystyle d={\overline {b}}}
Recurrence and difference relations [ edit ] The sequence of continuous Hahn polynomials satisfies the recurrence relation[ 4]
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{\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),}
where
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{\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}}}
The continuous Hahn polynomials are given by the Rodrigues-like formula[ 5]
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{\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}}}
Generating functions [ edit ] The continuous Hahn polynomials have the following generating function:[ 6]
∑
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{\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}}}
A second, distinct generating function is given by
∑
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{\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).}
Relation to other polynomials [ edit ] The Wilson polynomials are a generalization of the continuous Hahn polynomials.
The Bateman polynomials F n a =b =c =d =1/2 of the continuous Hahn polynomials by
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{\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).}
The Jacobi polynomials P n (α,β) (x) can be obtained as a limiting case of the continuous Hahn polynomials:[ 7]
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{\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).}
↑ 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 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
