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

$p_n(x;a,b,c,d)=i^n\frac{(a+c{)}_n(a+d{)}_n}{n!}{}_3{F}_2(\begin{array}{c}-n,n+a+b+c+d-1,a+ix\\ a+c,a+d\end{array};1)$

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.

Orthogonality

The continuous Hahn polynomials p_n(x;a,b,c,d) are orthogonal with respect to the weight function

$w(x)=\mathrm{\Gamma }(a+ix)\mathrm{\Gamma }(b+ix)\mathrm{\Gamma }(c-ix)\mathrm{\Gamma }(d-ix).$

In particular, they satisfy the orthogonality relation

$\begin{array}{rl}& \frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{\Gamma }(a+ix)\mathrm{\Gamma }(b+ix)\mathrm{\Gamma }(c-ix)\mathrm{\Gamma }(d-ix)p_m(x;a,b,c,d)p_n(x;a,b,c,d)dx\\ & =\frac{\mathrm{\Gamma }(n+a+c)\mathrm{\Gamma }(n+a+d)\mathrm{\Gamma }(n+b+c)\mathrm{\Gamma }(n+b+d)}{n!(2n+a+b+c+d-1)\mathrm{\Gamma }(n+a+b+c+d-1)}{\delta }_{nm}\end{array}$

for $\mathrm{\Re }(a)>0$, $\mathrm{\Re }(b)>0$, $\mathrm{\Re }(c)>0$, $\mathrm{\Re }(d)>0$, $c=\bar{a}$, $d=\bar{b}$.

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relation

$x{p}_n(x)=p_{n+1}(x)+i(A_n+C_n)p_n(x)-A_{n-1}{C}_n{p}_{n-1}(x),$

$\begin{array}{rl}\text{where}& p_n(x)=\frac{n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}{p}_n(x;a,b,c,d),\\ & A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)},\\ \text{and}& C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}.\end{array}$

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formula

$\begin{array}{rl}& \mathrm{\Gamma }(a+ix)\mathrm{\Gamma }(b+ix)\mathrm{\Gamma }(c-ix)\mathrm{\Gamma }(d-ix)p_n(x;a,b,c,d)\\ & =\frac{(-1{)}^n}{n!}\frac{d^n}{d{x}^n}\left(\mathrm{\Gamma }(a+\frac{n}{2}+ix)\mathrm{\Gamma }(b+\frac{n}{2}+ix)\mathrm{\Gamma }(c+\frac{n}{2}-ix)\mathrm{\Gamma }(d+\frac{n}{2}-ix)\right).\end{array}$

Generating functions

The continuous Hahn polynomials have the following generating function:

$\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(n+a+b+c+d)\mathrm{\Gamma }(a+c+1)\mathrm{\Gamma }(a+d+1)}{\mathrm{\Gamma }(a+b+c+d)\mathrm{\Gamma }(n+a+c+1)\mathrm{\Gamma }(n+a+d+1)}(-it{)}^n{p}_n(x;a,b,c,d)\\ & =(1-t{)}^{1-a-b-c-d}{}_3{F}_2(\begin{array}{c}\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix\\ a+c,a+d\end{array};-\frac{4t}{(1-t{)}^2}).\end{array}$

A second, distinct generating function is given by

$\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(a+c+1)\mathrm{\Gamma }(b+d+1)}{\mathrm{\Gamma }(n+a+c+1)\mathrm{\Gamma }(n+b+d+1)}{t}^n{p}_n(x;a,b,c,d)={}_1{F}_1(\begin{array}{c}a+ix\\ a+c\end{array};-it){}_1{F}_1(\begin{array}{c}d-ix\\ b+d\end{array};it).$

Relation to other polynomials

• The Wilson polynomials are a generalization of the continuous Hahn polynomials.
• The Bateman polynomials F_n(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by

$p_n(x;\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})=i^{n}n!F_n\left(2ix\right).$

• The Jacobi polynomials P_n^(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:

$P_n^{(\alpha ,\beta )}=\underset{t\to \mathrm{\infty }}{\lim }{t}^{-n}{p}_n(\frac{1}{2}xt;\frac{1}{2}(\alpha +1-it),\frac{1}{2}(\beta +1+it),\frac{1}{2}(\alpha +1+it),\frac{1}{2}(\beta +1-it)).$

References

• 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