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
:$p_n(x;a,b,c,d)= i^n\frac_3F_2\left( \begin -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end ; 1\right)$

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 relation
:$\begin&\frac\int_^\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_m(x;a,b,c,d)\,p_n(x;a,b,c,d)\,dx\\ &\qquad\qquad=\frac\,\delta_\end$
for $\Re(a)>0$, $\Re(b)>0$, $\Re(c)>0$, $\Re(d)>0$, $c = \overline$, $d = \overline$.

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relation
:$xp_n(x)=p_(x)+i(A_n+C_n)p_(x)-A_C_n p_(x),$
:$\begin \text\quad&p_n(x)=\fracp_n(x;a,b,c,d),\\ &A_n=-\frac,\\ \text\quad&C_n=\frac. \end$

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formula
:$\begin&\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_n(x;a,b,c,d)\\ &\qquad=\frac\frac\left(\Gamma\left(a+\frac+ix\right)\,\Gamma\left(b+\frac+ix\right)\,\Gamma\left(c+\frac-ix\right)\,\Gamma\left(d+\frac-ix\right)\right).\end$

Generating functions

The continuous Hahn polynomials have the following generating function:
:$\begin&\sum_^\frac(-it)^n p_n(x;a,b,c,d)\\ &\qquad=(1-t)^_3F_2\left( \begin \frac12(a+b+c+d-1), \frac12(a+b+c+d), a+ix\\ a+c, a+d\end ; -\frac \right).\end$
A second, distinct generating function is given by
:$\sum_^\fract^n p_n(x;a,b,c,d)=\,_1F_1\left( \begin a + ix \\ a + c\end ; -it\right)\,_1F_1\left( \begin d - ix \\ b + d\end ; it\right).$

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.
:$P_n^=\lim_t^p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right).$

References

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*{{Citation , last1=Andrews , first1=George E. , last2=Askey , first2 = Richard , last3=Roy , first3=Ranjan , title=Special functions , publisher=Cambridge University Press , location=Cambridge , series=Encyclopedia of Mathematics and its Applications 71 , isbn=978-0-521-62321-6 , year=1999
Special hypergeometric functions
Orthogonal polynomials