Hahn polynomials

Orthogonality

Hahn polynomials satisfy the orthogonality relations^{[citation needed]}

${\displaystyle \sum _{x=0}^{N-1}Q_{n}(x)Q_{m}(x)\rho (x)={\frac {1}{\pi _{n}}}\delta _{m,n},\quad \sum _{n=0}^{N-1}Q_{n}(x)Q_{n}(y)\pi _{n}={\frac {1}{\rho (x)}}\delta _{x,y},}$

where ${\displaystyle \delta _{x,y}}$ is the Kronecker delta and the weight functions are

${\displaystyle \rho (x;\alpha$ ;\beta ,N)={\binom {\alpha +x}{x}}{\binom {\beta +N-1-x}{N-1-x}}{\bigg /}{\binom {N+\alpha +\beta }{N-1}}}

and

${\displaystyle \pi _{n}(\alpha ,\beta ,N)={\binom {N-1}{n}}{\frac {2n+\alpha +\beta +1}{\alpha +\beta +1}}{\frac {\Gamma (\beta +1,n+\alpha +1,n+\alpha +\beta +1)}{\Gamma (\alpha +1,\alpha +\beta +1,n+\beta +1,n+1)}}{\bigg /}{\binom {N+\alpha +\beta +n}{n}}.}$

Orthogonality

The dual Hahn polynomials satisfy the orthogonality condition

${\displaystyle \sum _{s=a}^{b-1}w_{n}^{(c)}(s,a,b)w_{m}^{(c)}(s,a,b)\rho (s)\Delta x{\Big (}s-{\frac {1}{2}}{\Big )}=\delta _{nm}d_{n}^{2}}$

for ${\displaystyle n,m=0,1,...,N-1}$, where ${\displaystyle \Delta x(s)=x(s+1)-x(s)}$,

${\displaystyle \rho (s)={\frac {\Gamma (a+s+1)\Gamma (c+s+1)}{\Gamma (s-a+1)\Gamma (b-s)\Gamma (b+s+1)\Gamma (s-c+1)}}}$

and

${\displaystyle d_{n}^{2}={\frac {\Gamma (a+c+n+a)}{n!(b-a-n-1)!\Gamma (b-c-n)}}.}$

Numerical instability

As ${\displaystyle n}$ increases, the values that the discrete polynomials attain increase. As a result, obtaining numerical stability in calculating the polynomials requires the use of renormalized dual Hahn polynomials, defined as

${\displaystyle {\hat {w}}_{n}^{(c)}(s,a,b)=w_{n}^{(c)}(s,a,b){\sqrt {{\frac {\rho (s)}{d_{n}^{2}}}\Delta x{\Big (}s-{\frac {1}{2}}{\Big )}}}}$

for ${\displaystyle n=0,1,...,N-1}$. Here the orthogonality condition becomes

${\displaystyle \sum _{s=a}^{b-1}{\hat {w}}_{n}^{(c)}(s,a,b){\hat {w}}_{m}^{(c)}(s,a,b)=\delta _{m,n}}$

for ${\displaystyle n,m=0,1,...,N-1.}$

Orthogonality

The continuous Hahn polynomials are orthogonal with respect to the weight function

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

In particular, they satisfy the orthogonality relation^[5][6][7]

${\displaystyle {\begin{aligned}&{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{m}(x;a,b,c,d)\,p_{n}(x;a,b,c,d)\,dx\\&\qquad \qquad ={\frac {\Gamma (n+a+c)\,\Gamma (n+a+d)\,\Gamma (n+b+c)\,\Gamma (n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma (n+a+b+c+d-1)}}\,\delta _{nm}\end{aligned}}}$

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

Recurrence and difference relations

The sequence of continuous Hahn polynomials satisfies the recurrence relation^[5]

${\displaystyle (a+ix){\tilde {p}}_{n}(x)=A_{n}{\tilde {p}}_{n+1}(x)-(A_{n}+C_{n}){\tilde {p}}_{n}(x)+C_{n}{\tilde {p}}_{n-1}(x),}$

where

${\displaystyle {\begin{aligned}&{\tilde {p}}_{n}(x)={\tilde {p}}_{n}(x;a,b,c,d)={\frac {n!}{i^{n}(a+c)_{n}(a+d)_{n}}}p_{n}(x;a,b,c,d),\\[6pt]&A_{n}={\frac {-(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\end{aligned}}}$

and

${\displaystyle C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.}$

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-type formula^[8]

${\displaystyle {\begin{aligned}&\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{n}(x;a,b,c,d)\\[5pt]&\quad ={\frac {(-1)^{n}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(\Gamma \left(a+{\frac {n}{2}}+ix\right)\,\Gamma \left(b+{\frac {n}{2}}+ix\right)\,\Gamma \left(c+{\frac {n}{2}}-ix\right)\,\Gamma \left(d+{\frac {n}{2}}-ix\right)\right).\end{aligned}}}$

Generating functions

The continuous Hahn polynomials have the following generating function:^[8]

${\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }{\frac {\Gamma (n+a+b+c+d)\,\Gamma (a+c+1)\,\Gamma (a+d+1)}{\Gamma (a+b+c+d)\,\Gamma (n+a+c+1)\,\Gamma (n+a+d+1)}}(-it)^{n}p_{n}(x;a,b,c,d)\\[4pt]&\qquad =(1-t)^{1-a-b-c-d}{}_{3}F_{2}\left({\begin{array}{c}{\frac {1}{2}}(a+b+c+d-1),{\frac {1}{2}}(a+b+c+d),a+ix\\a+c,a+d\end{array}};-{\frac {4t}{(1-t)^{2}}}\right).\end{aligned}}}$

A second, distinct generating function is given by

${\displaystyle \sum _{n=0}^{\infty }{\frac {\Gamma (a+c+1)\,\Gamma (b+d+1)}{\Gamma (n+a+c+1)\,\Gamma (n+b+d+1)}}t^{n}p_{n}(x;a,b,c,d)=\,_{1}F_{1}\left({\begin{array}{c}a+ix\\a+c\end{array}};-it\right)\,_{1}F_{1}\left({\begin{array}{c}d-ix\\b+d\end{array}};it\right).}$

Relation of continuous Hahn polynomials 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

${\displaystyle p_{n}\left(x;{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}}\right)=i^{n}n!F_{n}\left(2ix\right).}$

• The Jacobi polynomials P_n^(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:^[9]

${\displaystyle P_{n}^{(\alpha ,\beta )}=\lim _{t\to \infty }t^{-n}p_{n}\left({\tfrac {1}{2}}xt;{\tfrac {1}{2}}(\alpha +1-it),{\tfrac {1}{2}}(\beta +1+it),{\tfrac {1}{2}}(\alpha +1+it),{\tfrac {1}{2}}(\beta +1-it)\right).}$

Relation of continuous dual Hahn polynomials to other polynomials

• Wilson polynomials are a generalization of continuous dual Hahn polynomials

Definition

The polynomials are given in terms of basic hypergeometric functions by

${\displaystyle Q_{n}(q^{-x};a,b,N;q)={}_{3}\phi _{2}\left[{\begin{matrix}q^{-n},abq^{n+1},q^{-x}\\aq,q^{-N}\end{matrix}};q,q\right].}$

Dual q-Hahn polynomials

In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Continuous q-Hahn polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Gallery

Relation of q-Hahn polynomials to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

${\displaystyle \lim _{a\to \infty }Q_{n}(q^{-x};a;p,N|q)=K_{n}^{qtm}(q^{-x};p,N;q)}$

q-Hahn polynomials→ Hahn polynomials

make the substitution${\displaystyle \alpha =q^{\alpha }}$,${\displaystyle \beta =q^{\beta }}$ into definition of q-Hahn polynomials, and find the limit q→1, we obtain

${\displaystyle {}_{3}F_{2}(-n,\alpha +\beta +n+1,-x,\alpha +1,-N,1)}$，which is exactly Hahn polynomials.

Continuous dual q-Hahn polynomials

In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Gallery