Continuous Hahn polynomials
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
- [math]\displaystyle{ p_n(x;a,b,c,d)= i^n\frac{(a+c)_n(a+d)_n}{n!}{}_3F_2\left( \begin{array}{c} -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end{array} ; 1\right) }[/math]
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
- [math]\displaystyle{ w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix). }[/math]
In particular, they satisfy the orthogonality relation[1][2][3]
- [math]\displaystyle{ \begin{align}&\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_{n m}\end{align} }[/math]
for [math]\displaystyle{ \Re(a)\gt 0 }[/math], [math]\displaystyle{ \Re(b)\gt 0 }[/math], [math]\displaystyle{ \Re(c)\gt 0 }[/math], [math]\displaystyle{ \Re(d)\gt 0 }[/math], [math]\displaystyle{ c = \overline{a} }[/math], [math]\displaystyle{ d = \overline{b} }[/math].
Recurrence and difference relations
The sequence of continuous Hahn polynomials satisfies the recurrence relation[4]
- [math]\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), }[/math]
- [math]\displaystyle{ \begin{align} \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{align} }[/math]
Rodrigues formula
The continuous Hahn polynomials are given by the Rodrigues-like formula[5]
- [math]\displaystyle{ \begin{align}&\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{align} }[/math]
Generating functions
The continuous Hahn polynomials have the following generating function:[6]
- [math]\displaystyle{ \begin{align}&\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}{}_3F_2\left( \begin{array}{c} \frac12(a+b+c+d-1), \frac12(a+b+c+d), a+ix\\ a+c, a+d\end{array} ; -\frac{4t}{(1-t)^2} \right).\end{align} }[/math]
A second, distinct generating function is given by
- [math]\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)=\,_1F_1\left( \begin{array}{c} a + ix \\ a + c\end{array} ; -it\right)\,_1F_1\left( \begin{array}{c} d - ix \\ b + d\end{array} ; it\right). }[/math]
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
- [math]\displaystyle{ p_n\left(x;\tfrac12,\tfrac12,\tfrac12,\tfrac12\right) = i^n n!F_n\left(2ix\right). }[/math]
- The Jacobi polynomials Pn(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:[7]
- [math]\displaystyle{ P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right). }[/math]
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
- 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
- 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18.19
- 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
Original source: https://en.wikipedia.org/wiki/Continuous Hahn polynomials.
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