Hahn polynomials

In mathematics, Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev^[1] and rediscovered by Wolfgang Hahn.^[2] The Hahn class is a name for special cases of Hahn polynomials, such as Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.

Hahn polynomials are defined in terms of generalized hypergeometric functions by^[3]

${\displaystyle Q_n(x;\alpha ,\beta ,N)={}_3{F}_2(-n,-x,n+\alpha +\beta +1;\alpha +1,-N+1;1)}$

for ${\displaystyle n=0,1,\dots ,N}$. If ${\displaystyle \alpha =\beta =0}$, these polynomials are identical to the discrete Chebyshev polynomials up to a scale factor.

Hahn polynomials have dual, continuous and continuous dual versions. These polynomials all have q-analogs. Hahn polynomials are a limiting case of Racah polynomials.^[4]

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

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

${\displaystyle \rho (x;\alpha ;\beta ,N)={\displaystyle (\genfrac{}{}{0ex}{}{\alpha +x}{x})}{\displaystyle (\genfrac{}{}{0ex}{}{\beta +N-1-x}{N-1-x})}/{\displaystyle (\genfrac{}{}{0ex}{}{N+\alpha +\beta }{N-1})}}$

and

${\displaystyle {\pi }_n(\alpha ,\beta ,N)={\displaystyle (\genfrac{}{}{0ex}{}{N-1}{n})}\frac{2n+\alpha +\beta +1}{\alpha +\beta +1}\frac{\mathrm{\Gamma }(\beta +1,n+\alpha +1,n+\alpha +\beta +1)}{\mathrm{\Gamma }(\alpha +1,\alpha +\beta +1,n+\beta +1,n+1)}/{\displaystyle (\genfrac{}{}{0ex}{}{N+\alpha +\beta +n}{n})}.}$

${\displaystyle w_n^{(c)}(s,a,b)=\frac{(a-b+1{)}_n(a+c+1{)}_n}{n!}\cdot {}_3{F}_2(\begin{array}{c}-na-sa+s+1\\ -ba+c+1\end{array};1)}$

for ${\displaystyle n=0,1,...,N-1}$, and the parameters ${\displaystyle a,b,c}$ are restricted to

${\displaystyle -\frac{1}{2}<a<b,|c|<1+a,b=a+N.}$

Here ${\displaystyle (u{)}_k}$ is the rising factorial, and ${\displaystyle {}_3{F}_2}$ is a generalized hypergeometric function.

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

The dual Hahn polynomials satisfy the orthogonality condition

${\displaystyle {\displaystyle \sum _{s=a}^{b-1}}{w}_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho (s)\mathrm{\Delta }x(s-\frac{1}{2})={\delta }_{nm}{d}_n^2}$

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

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

and

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

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

${\displaystyle {\hat{w}}_n^{(c)}(s,a,b)=w_n^{(c)}(s,a,b)\sqrt{\frac{\rho (s)}{d_n^2}\mathrm{\Delta }x(s-\frac{1}{2})}}$

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

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

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

Hahn polynomials have a continuous version, the continuous Hahn polynomials, which are also a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by

${\displaystyle p_n(x;a,b,c,d)=i^n\frac{(a+c{)}_n(a+d{)}_n}{n!}\cdot {}_3{F}_2(\begin{array}{c}-nn+a+b+c+d-1a+ix\\ a+ca+d\end{array};1)}$

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

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

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

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

${\displaystyle \begin{array}{rl}& \frac{1}{2\pi }{\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}}\mathrm{\Gamma }(a+ix)\mathrm{\Gamma }(b+ix)\mathrm{\Gamma }(c-ix)\mathrm{\Gamma }(d-ix)p_m(x;a,b,c,d)p_n(x;a,b,c,d)dx\\ & =\frac{\mathrm{\Gamma }(n+a+c)\mathrm{\Gamma }(n+a+d)\mathrm{\Gamma }(n+b+c)\mathrm{\Gamma }(n+b+d)}{n!(2n+a+b+c+d-1)\mathrm{\Gamma }(n+a+b+c+d-1)}{\delta }_{nm}\end{array}}$

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

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

${\displaystyle \begin{array}{rl}& {\tilde{p}}_n(x)={\tilde{p}}_n(x;a,b,c,d)=\frac{n!}{i^n(a+c{)}_n(a+d{)}_n}{p}_n(x;a,b,c,d),\\ & A_n=\frac{-(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)},\end{array}}$

and

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

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

${\displaystyle \begin{array}{rl}& \mathrm{\Gamma }(a+ix)\mathrm{\Gamma }(b+ix)\mathrm{\Gamma }(c-ix)\mathrm{\Gamma }(d-ix)p_n(x;a,b,c,d)\\ & =\frac{(-1{)}^n}{n!}\frac{d^n}{d{x}^n}\left(\mathrm{\Gamma }(a+\frac{n}{2}+ix)\mathrm{\Gamma }(b+\frac{n}{2}+ix)\mathrm{\Gamma }(c+\frac{n}{2}-ix)\mathrm{\Gamma }(d+\frac{n}{2}-ix)\right).\end{array}}$

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

${\displaystyle \begin{array}{rl}& {\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{\mathrm{\Gamma }(n+a+b+c+d)\mathrm{\Gamma }(a+c+1)\mathrm{\Gamma }(a+d+1)}{\mathrm{\Gamma }(a+b+c+d)\mathrm{\Gamma }(n+a+c+1)\mathrm{\Gamma }(n+a+d+1)}(-it{)}^n{p}_n(x;a,b,c,d)\\ & =(1-t{)}^{1-a-b-c-d}{}_3{F}_2(\begin{array}{c}\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix\\ a+c,a+d\end{array};-\frac{4t}{(1-t{)}^2}).\end{array}}$

A second, distinct generating function is given by

${\displaystyle {\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{\mathrm{\Gamma }(a+c+1)\mathrm{\Gamma }(b+d+1)}{\mathrm{\Gamma }(n+a+c+1)\mathrm{\Gamma }(n+b+d+1)}{t}^n{p}_n(x;a,b,c,d)={}_1{F}_1(\begin{array}{c}a+ix\\ a+c\end{array};-it){}_1{F}_1(\begin{array}{c}d-ix\\ b+d\end{array};it).}$

• 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

${\displaystyle p_n(x;\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})=i^{n}n!F_n\left(2ix\right).}$

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

${\displaystyle P_n^{(\alpha ,\beta )}=\underset{t\to \mathrm{\infty }}{\lim }{t}^{-n}{p}_n(\frac{1}{2}xt;\frac{1}{2}(\alpha +1-it),\frac{1}{2}(\beta +1+it),\frac{1}{2}(\alpha +1+it),\frac{1}{2}(\beta +1-it)).}$

In mathematics, the continuous dual 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

${\displaystyle S_n(x^2;a,b,c)={}_3{F}_2(-n,a+ix,a-ix;a+b,a+c;1).}$

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Closely related polynomials include the dual Hahn polynomials R_n(x;γ,δ,N), the continuous Hahn polynomials p_n(x,a,b, a, b), and the Hahn polynomials. 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.

• Wilson polynomials are a generalization of continuous dual Hahn polynomials

In mathematics, the 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.

The polynomials are given in terms of basic hypergeometric functions by

${\displaystyle Q_n(q^{-x};a,b,N;q)={}_3{\varphi }_2[\begin{array}{c}{q}^{-n},ab{q}^{n+1},q^{-x}\\ aq,q^{-N}\end{array};q,q].}$

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.

The polynomials are given in terms of basic hypergeometric functions. ${\displaystyle R_n(q^{-x}+\gamma \delta q^{x+1},\gamma ,\delta ,N|q)={}_3{\varphi }_2[\begin{array}{c}{q}^{-n},q^{-x},\gamma \delta q^{x+1}\\ \gamma q,q^{-N}\end{array};q,q],n=0,1,2,...,N}$

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.

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ^[10]

${\displaystyle p_n(x;a,b,c,d|q)=a^{-n}{e}^{-inu}(ab{e}^{2iu},ac,ad;q{)}_n{}_4{\varphi }_3(q^{-n},abcd{q}^{n-1},a{e}^{i(t+2u)},a{e}^{-it};ab{e}^{2iu},ac,ad;q;q)}$

${\displaystyle x=\cos (t+u)}$

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

${\displaystyle \underset{a\to \mathrm{\infty }}{\lim }{Q}_n(q^{-x};a;p,N|q)=K_n^{qtm}(q^{-x};p,N;q)}$

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.

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.

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ^[11]

${\displaystyle p_n(x;a,b,c\mid q)=\frac{(ab,ac;q{)}_n}{a^n}{}_3{\varphi }_2(q^{-n},a{e}^{i\theta },a{e}^{-i\theta };ab,ac\mid q;q)}$

In which ${\displaystyle x=\cos (\theta )}$

• Racah polynomials are a generalization of Hahn polynomials

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