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

: $p_n(x;a,b,c,d)= i^n\frac{(a+c)_n(a+d)_n}{n!}{}_3F_2\left( \begin{array}{c} -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end{array} ; 1\right)$

{{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=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

: $w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix).$

In particular, they satisfy the orthogonality relationKoekoek, 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.

: $\begin{align}&\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_m(x;a,b,c,d)\,p_n(x;a,b,c,d)\,dx\\
&\qquad\qquad=\frac{\Gamma(n+a+c)\,\Gamma(n+a+d)\,\Gamma(n+b+c)\,\Gamma(n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma(n+a+b+c+d-1)}\,\delta_{n m}\end{align}$

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

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relationKoekoek, Lesky, & Swarttouw (2010), p. 201.

: $xp_n(x)=p_{n+1}(x)+i(A_n+C_n)p_{n}(x)-A_{n-1}C_n p_{n-1}(x),$

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

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formulaKoekoek, Lesky, & Swarttouw (2010), p. 202.

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

Generating functions

The continuous Hahn polynomials have the following generating function:Koekoek, Lesky, & Swarttouw (2010), p. 202.

: $\begin{align}&\sum_{n=0}^{\infty}\frac{\Gamma(n+a+b+c+d)\,\Gamma(a+c+1)\,\Gamma(a+d+1)}{\Gamma(a+b+c+d)\,\Gamma(n+a+c+1)\,\Gamma(n+a+d+1)}(-it)^n p_n(x;a,b,c,d)\\
&\qquad=(1-t)^{1-a-b-c-d}{}_3F_2\left( \begin{array}{c} \frac12(a+b+c+d-1), \frac12(a+b+c+d), a+ix\\ a+c, a+d\end{array} ; -\frac{4t}{(1-t)^2} \right).\end{align}$

A second, distinct generating function is given by

: $\sum_{n=0}^{\infty}\frac{\Gamma(a+c+1)\,\Gamma(b+d+1)}{\Gamma(n+a+c+1)\,\Gamma(n+b+d+1)}t^n p_n(x;a,b,c,d)=\,_1F_1\left( \begin{array}{c} a + ix \\ a + c\end{array} ; -it\right)\,_1F_1\left( \begin{array}{c} d - ix \\ b + d\end{array} ; it\right).$

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

: $p_n\left(x;\tfrac12,\tfrac12,\tfrac12,\tfrac12\right) = i^n n!F_n\left(2ix\right).$

The Jacobi polynomials P_n^(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:Koekoek, Lesky, & Swarttouw (2010), p. 203.

: $P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right).$

References

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{{dlmf|id=18.19|title=Hahn Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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Category:Special hypergeometric functions

Category:Orthogonal polynomials