Continuous Hahn polynomials explained

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 ((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)

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^{[1] [2] [3]} \begin{align}& (1)/(2 \pi) \int ^{inf ty} _{-inf ty} \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\\ & = (\Gamma(n + a + c)\Gamma(n + a + d)\Gamma(n + b + c)\Gamma(n + b + d))/(n!( 2 n + a + b + c + d -1 )\Gamma(n + a + b + c + d -1 ))} \delta _{n} \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^[4]

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} where &p _{n(x) = (n!(n + a + b + c + d -1 )!)/(( 2 n + a + b + c + d -1 )!)} p _{n(x;a,b,c,d),\\ &A} ^{n =- ((n + a + b + c + d -1 )(n + a + c)(n + a + d))/(( 2 n + a + b + c + d -1 )( 2 n + a + b + c + d))} ,\\ and &C _{n = (n(n + b + c -1 )(n + b + d -1 ))/(( 2 n + a + b + c + d -2 )( 2 n + a + b + c + d -1 ))} . \end{align}

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formula^[5]

\begin{align}&\Gamma(a + ix)\Gamma(b + ix)\Gamma(c - ix)\Gamma(d - ix)p _{n(x;a,b,c,d)\\ & = (( -1 ) ^{n})/(n!)} (d ^{n})/(\left(\Gamma\left(a + dx ^{n}) (n)/(+ ix\right)\Gamma\left(b + 2) (n)/(+ ix\right)\Gamma\left(c + 2) (n)/(- ix\right)\Gamma\left(d + 2) (n)/(2) - ix\right)\right).\end{align}

Generating functions

The continuous Hahn polynomials have the following generating function:^[6]

\begin{align}&\sum ^{inf ty} _{n =0} (\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 ^{1- a - b - c - d} _{n(x;a,b,c,d)\\ & = ( 1- t)} {} _{3F} _{2\left(} \begin{array}{c} (12 (a + b + c + d -1 ),)/((12 (a + b + c + d),)/(a + ix\ a + c,)) a + d\end{array} ; - (4 t)/(( 1- t) ^{2}) \right).\end{align} A second, distinct generating function is given by\sum ^{inf ty} _{n =0} (\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:^[7]

P ^{(\alpha,\beta)} _{n} = \lim _{t\toinfty} t ^{-n} p _{n\left(\tfrac12xt;} \tfrac 12 (\alpha +1- it), \tfrac 12 (\beta +1+ it), \tfrac 12 (\alpha +1+ it), \tfrac 12 (\beta +1- it)\right).

Notes and 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.