Continuous Hahn polynomials

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 relation :\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 relation :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 formula :\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: :\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: :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

References

1. Koekoek, Lesky, & Swarttouw (2010), p. 200.
2. Askey, R. (1985), "Continuous Hahn polynomials", ''J. Phys. A: Math. Gen.'' '''18''': pp. L1017-L1019.
3. Andrews, Askey, & Roy (1999), p. 333.
4. Koekoek, Lesky, & Swarttouw (2010), p. 201.
5. Koekoek, Lesky, & Swarttouw (2010), p. 202.
6. Koekoek, Lesky, & Swarttouw (2010), p. 202.
7. Koekoek, Lesky, & Swarttouw (2010), p. 203.