Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) ${\displaystyle P_{n}^{(\alpha ,\beta )}(x)}$ are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ${\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}$ on the interval ${\displaystyle [-1,1]}$. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.^[1]

The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

The Jacobi polynomials are defined via the hypergeometric function as follows:^{[2][1]: IV.1}

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\tfrac {1}{2}}(1-z)\right),}$

where ${\displaystyle (\alpha +1)_{n}}$ is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\,\Gamma (\alpha +\beta +n+1)}}\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m}.}$

An equivalent definition is given by Rodrigues' formula:^{[1]: IV.3 [3]}

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(-1)^{n}}{2^{n}n!}}(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n}}{dz^{n}}}\left\{(1-z)^{\alpha }(1+z)^{\beta }\left(1-z^{2}\right)^{n}\right\}.}$

If ${\displaystyle \alpha =\beta =0}$, then it reduces to the Legendre polynomials:

${\displaystyle P_{n}(z)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dz^{n}}}(z^{2}-1)^{n}\;.}$

The Jacobi polynomials ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ is, up to scaling, the unique polynomial solution of the Sturm–Liouville problem^[1]: IV.2

${\displaystyle \left(1-x^{2}\right)y''+(\beta -\alpha -(\alpha +\beta +2)x)y'=\lambda y}$

where ${\displaystyle \lambda =-n(n+\alpha +\beta +1)}$. The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial solutions to Sturm–Liouville problems with polynomial coefficients.

For real ${\displaystyle x}$ the Jacobi polynomial can alternatively be written as

${\displaystyle P_{n}^{(\alpha ,\beta )}(x)=\sum _{s=0}^{n}{n+\alpha \choose n-s}{n+\beta \choose s}\left({\frac {x-1}{2}}\right)^{s}\left({\frac {x+1}{2}}\right)^{n-s}}$

and for integer ${\displaystyle n}$

${\displaystyle {z \choose n}={\begin{cases}{\frac {\Gamma (z+1)}{\Gamma (n+1)\Gamma (z-n+1)}}&n\geq 0\\0&n<0\end{cases}}}$

where ${\displaystyle \Gamma (z)}$ is the gamma function.

In the special case that the four quantities ${\displaystyle n}$, ${\displaystyle n+\alpha }$, ${\displaystyle n+\beta }$, ${\displaystyle n+\alpha +\beta }$ are nonnegative integers, the Jacobi polynomial can be written as

${\displaystyle P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\sum _{s=0}^{n}{\frac {1}{s!(n+\alpha -s)!(\beta +s)!(n-s)!}}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.}$

1

The sum extends over all integer values of ${\displaystyle s}$ for which the arguments of the factorials are nonnegative.

${\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1,}$

${\displaystyle P_{1}^{(\alpha ,\beta )}(z)=(\alpha +1)+(\alpha +\beta +2){\frac {z-1}{2}},}$

${\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)(\alpha +2)}{2}}+(\alpha +2)(\alpha +\beta +3){\frac {z-1}{2}}+{\frac {(\alpha +\beta +3)(\alpha +\beta +4)}{2}}\left({\frac {z-1}{2}}\right)^{2}.}$

$${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (1+2n+\alpha +\beta )}{\Gamma (1+n)\Gamma (1+n+\alpha +\beta )}}\left({\frac {z}{2}}\right)^{n}+{\text{ lower-degree terms }}}$$Thus, the leading coefficient is ${\displaystyle {\frac {\Gamma (1+2n+\alpha +\beta )}{2^{n}n!\Gamma (1+n+\alpha +\beta )}}}$.

The Jacobi polynomials satisfy the orthogonality condition

${\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\,dx={\frac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{nm},\qquad \alpha ,\ \beta >-1.}$

As defined, they do not have unit norm with respect to the weight. This can be corrected by dividing by the square root of the right hand side of the equation above, when ${\displaystyle n=m}$.

Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:

${\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n}.}$

The polynomials have the symmetry relation

${\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z);}$

thus the other terminal value is

${\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n}.}$

The ${\displaystyle k}$th derivative of the explicit expression leads to

${\displaystyle {\frac {d^{k}}{dz^{k}}}P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)}}P_{n-k}^{(\alpha +k,\beta +k)}(z).}$

The 3-term recurrence relation for the Jacobi polynomials of fixed ${\displaystyle \alpha }$, ${\displaystyle \beta }$ is:^[1]: IV.5

${\displaystyle {\begin{aligned}&2n(n+\alpha +\beta )(2n+\alpha +\beta -2)P_{n}^{(\alpha ,\beta )}(z)\\&\qquad =(2n+\alpha +\beta -1){\Big \{}(2n+\alpha +\beta )(2n+\alpha +\beta -2)z+\alpha ^{2}-\beta ^{2}{\Big \}}P_{n-1}^{(\alpha ,\beta )}(z)-2(n+\alpha -1)(n+\beta -1)(2n+\alpha +\beta )P_{n-2}^{(\alpha ,\beta )}(z),\end{aligned}}}$

for ${\displaystyle n=2,3,\ldots }$. Writing for brevity ${\displaystyle a:=n+\alpha }$, ${\displaystyle b:=n+\beta }$ and ${\displaystyle c:=a+b=2n+\alpha +\beta }$, this becomes in terms of ${\displaystyle a,b,c}$

${\displaystyle 2n(c-n)(c-2)P_{n}^{(\alpha ,\beta )}(z)=(c-1){\Big \{}c(c-2)z+(a-b)(c-2n){\Big \}}P_{n-1}^{(\alpha ,\beta )}(z)-2(a-1)(b-1)c\;P_{n-2}^{(\alpha ,\beta )}(z).}$

Since the Jacobi polynomials can be described in terms of the hypergeometric function, recurrences of the hypergeometric function give equivalent recurrences of the Jacobi polynomials. In particular, Gauss' contiguous relations correspond to the identities^{[4]: Appx.B}

${\displaystyle {\begin{aligned}(z-1){\frac {d}{dz}}P_{n}^{(\alpha ,\beta )}(z)&={\frac {1}{2}}(z-1)(1+\alpha +\beta +n)P_{n-1}^{(\alpha +1,\beta +1)}\\&=nP_{n}^{(\alpha ,\beta )}-(\alpha +n)P_{n-1}^{(\alpha ,\beta +1)}\\&=(1+\alpha +\beta +n)\left(P_{n}^{(\alpha ,\beta +1)}-P_{n}^{(\alpha ,\beta )}\right)\\&=(\alpha +n)P_{n}^{(\alpha -1,\beta +1)}-\alpha P_{n}^{(\alpha ,\beta )}\\&={\frac {2(n+1)P_{n+1}^{(\alpha ,\beta -1)}-\left(z(1+\alpha +\beta +n)+\alpha +1+n-\beta \right)P_{n}^{(\alpha ,\beta )}}{1+z}}\\&={\frac {(2\beta +n+nz)P_{n}^{(\alpha ,\beta )}-2(\beta +n)P_{n}^{(\alpha ,\beta -1)}}{1+z}}\\&={\frac {1-z}{1+z}}\left(\beta P_{n}^{(\alpha ,\beta )}-(\beta +n)P_{n}^{(\alpha +1,\beta -1)}\right)\,.\end{aligned}}}$

The generating function of the Jacobi polynomials is given by

${\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(z)t^{n}=2^{\alpha +\beta }R^{-1}(1-t+R)^{-\alpha }(1+t+R)^{-\beta },}$

where

${\displaystyle R=R(z,t)=\left(1-2zt+t^{2}\right)^{\frac {1}{2}}~,}$

and the branch of square root is chosen so that ${\displaystyle R(z,0)=1}$.^[1]: IV.4

The Jacobi polynomials reduce to other classical polynomials.^[5]

Ultraspherical:$${\displaystyle {\begin{aligned}C_{n}^{(\lambda )}(x)&={\frac {(2\lambda )_{n}}{\left(\lambda +{\frac {1}{2}}\right)_{n}}}P_{n}^{\left(\lambda -{\frac {1}{2}},\lambda -{\frac {1}{2}}\right)}(x),\\P_{n}^{(\alpha ,\alpha )}(x)&={\frac {(\alpha +1)_{n}}{(2\alpha +1)_{n}}}C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(x).\end{aligned}}}$$Legendre:$${\displaystyle P_{n}(x)=C_{n}^{\left({\frac {1}{2}}\right)}(x)=P_{n}^{(0,0)}(x)}$$Chebyshev:$${\displaystyle {\begin{aligned}T_{n}(x)&=P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(x)/P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(1),\\U_{n}(x)&=C_{n}^{(1)}(x)=(n+1)P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(x)/P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(1),\\V_{n}(x)&=P_{n}^{\left(-{\frac {1}{2}},{\frac {1}{2}}\right)}(x)/P_{n}^{\left(-{\frac {1}{2}},{\frac {1}{2}}\right)}(1),\\W_{n}(x)&=(2n+1)P_{n}^{\left({\frac {1}{2}},-{\frac {1}{2}}\right)}(x)/P_{n}^{\left({\frac {1}{2}},-{\frac {1}{2}}\right)}(1).\\T_{n}^{*}(x)&=T_{n}(2x-1),\\U_{n}^{*}(x)&=U_{n}(2x-1).\end{aligned}}}$$Laguerre:$${\displaystyle {\begin{aligned}\lim _{\beta \rightarrow \infty }P_{n}^{(\alpha ,\beta )}(1-(2x/\beta ))&=L_{n}^{(\alpha )}(x).\\\lim _{\alpha \rightarrow \infty }P_{n}^{(\alpha ,\beta )}((2x/\alpha )-1)&=(-1)^{n}L_{n}^{(\beta )}(x).\end{aligned}}}$$Hermite:$${\displaystyle \lim _{\alpha \rightarrow \infty }\alpha ^{-{\frac {1}{2}}n}P_{n}^{(\alpha ,\alpha )}\left(\alpha ^{-{\frac {1}{2}}}x\right)={\frac {H_{n}(x)}{2^{n}n!}}}$$

The Jacobi polynomials appear as the eigenfunctions of the Markov process on ${\displaystyle [-1,+1]}$$${\displaystyle {\mathcal {L}}=\left(1-x^{2}\right){\frac {\partial ^{2}}{\partial ^{2}x}}+(px+q){\frac {\partial }{\partial x}}}$$defined up to the time it hits the boundary. For ${\displaystyle p=-(\beta +\alpha +2),q=\beta -\alpha }$, we have$${\displaystyle {\mathcal {L}}P_{n}^{(\alpha ,\beta )}=-n(n+\alpha +\beta +1)P_{n}^{(\alpha ,\beta )}}$$Thus this process is named the Jacobi process.^[6][7]

Let

• ${\displaystyle J^{(\alpha ,\beta )}:=-\left(1-x^{2}\right){\frac {d^{2}}{dx^{2}}}-[\beta -\alpha -(\alpha +\beta +2)x]{\frac {d}{dx}}}$
• ${\displaystyle T_{t}^{(\alpha ,\beta )}:=e^{-tJ^{(\alpha ,\beta )}}}$
• ${\displaystyle h_{n}^{(\alpha ,\beta )}=\int _{-1}^{1}\left[P_{n}^{(\alpha ,\beta )}(x)\right]^{2}(1-x)^{\alpha }(1+x)^{\beta }dx={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{(2n+\alpha +\beta +1)\Gamma (n+\alpha +\beta +1)\Gamma (n+1)}}}$
• ${\displaystyle G_{t}^{(\alpha ,\beta )}(x,y)=\sum _{n=0}^{\infty }\exp(-tn(n+\alpha +\beta +1)){\frac {P_{n}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(y)}{h_{n}^{(\alpha ,\beta )}}},\quad x,y\in [-1,1],\quad t>0,}$
• ${\displaystyle d\rho _{(\alpha ,\beta )}(x)=(1-x)^{\alpha }(1+x)^{\beta }dx}$

Then, for any ${\displaystyle f\in L^{1}\left(d\rho _{(\alpha ,\beta )}\right)}$,^[8]$${\displaystyle T_{t}^{(\alpha ,\beta )}f(x)=\int _{-1}^{1}G_{t}^{(\alpha ,\beta )}(x,y)f(y)d\varrho _{(\alpha ,\beta )}(y)}$$Thus, ${\displaystyle G_{t}^{(\alpha ,\beta )}}$ is called the Jacobi heat kernel.

The discriminant is^[9]$${\displaystyle \operatorname {Disc} \left(P_{n}^{(\alpha ,\beta )}\right)=2^{-n(n-1)}\prod _{j=1}^{n}j^{j-2n+2}(j+\alpha )^{j-1}(j+\beta )^{j-1}(n+j+\alpha +\beta )^{n-j}}$$Bailey’s formula:^[8][10]$${\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }{\frac {P_{n}^{(\alpha ,\beta )}(\cos \theta )P_{n}^{(\alpha ,\beta )}(\cos \varphi )}{h_{n}^{(\alpha ,\beta )}}}r^{n}={\frac {\Gamma (\alpha +\beta +2)}{2^{\alpha +\beta +1}\Gamma (\alpha +1)\Gamma (\beta +1)}}{\frac {1-r}{(1+r)^{\alpha +\beta +2}}}\\&\quad \times F_{4}\left({\frac {\alpha +\beta +2}{2}},{\frac {\alpha +\beta +3}{2}};\alpha +1,\beta +1;\left({\frac {2\sin {\frac {\theta }{2}}\sin {\frac {\varphi }{2}}}{r^{1/2}+r^{-1/2}}}\right)^{2},\left({\frac {2\cos {\frac {\theta }{2}}\cos {\frac {\varphi }{2}}}{r^{1/2}+r^{-1/2}}}\right)^{2}\right)\end{aligned}}}$$where ${\displaystyle |r|<1,\alpha ,\beta >-1}$, and ${\displaystyle F_{4}}$ is Appel's hypergeometric function of two variables. This is an analog of the Mehler kernel for Hermite polynomials, and the Hardy–Hille formula for Laguerre polynomials.

Laplace-type integral representation:^[11]$${\displaystyle {\begin{aligned}P_{n}^{\left(\alpha ,\beta \right)}\left(1-2t^{2}\right)=&{\frac {(-1)^{n}2^{2n}}{\pi (2n)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma \left(\alpha +{\frac {1}{2}}\right)\Gamma \left(\beta +{\frac {1}{2}}\right)}}.\\&\int _{-1}^{1}\int _{-1}^{1}\left(tu\pm i{\sqrt {1-t^{2}}}v\right)^{2n}\left(1-u^{2}\right)^{\alpha -{\frac {1}{2}}}\left(1-v^{2}\right)^{\beta -{\frac {1}{2}}}dudv.\end{aligned}}}$$

If ${\displaystyle \alpha ,\beta >-1}$, then ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ has ${\displaystyle n}$ real roots. Thus in this section we assume ${\displaystyle \alpha ,\beta >-1}$ by default. This section is based on.^[12][13]

Define:

• ${\displaystyle j_{\alpha ,m}}$ are the positive zero of the Bessel function of the first kind ${\displaystyle J_{\alpha }}$, ordered such that ${\displaystyle 0<j_{\alpha ,1}<j_{\alpha ,2}<\cdots }$.
• ${\displaystyle \theta _{n,m}=\theta _{n,m}^{(\alpha ,\beta )}}$ are the zeroes of ${\displaystyle P_{n}^{(\alpha ,\beta )}\left(\cos \theta \right)}$, ordered such that ${\displaystyle 0<\theta _{n,1}<\theta _{n,2}<\cdots <\theta _{n,n}<\pi }$.
• ${\displaystyle \rho =n+{\frac {1}{2}}(\alpha +\beta +1)}$
• ${\displaystyle \phi _{m}=j_{\alpha ,m}/\rho }$

${\displaystyle \theta _{n,m}}$ is strictly monotonically increasing with ${\displaystyle \alpha }$ and strictly monotonically decreasing with ${\displaystyle \beta }$.^[12]

If ${\displaystyle \alpha =\beta }$, and ${\displaystyle m<n/2}$, then ${\displaystyle \theta _{n,m}}$ is strictly monotonically increasing with ${\displaystyle \alpha }$.^[12]

When ${\displaystyle \alpha ,\beta \in [-1/2,+1/2]}$,^[12]

• ${\displaystyle \theta _{n,m}^{(-{\frac {1}{2}},{\frac {1}{2}})}={\frac {(m-{\tfrac {1}{2}})\pi }{n+{\tfrac {1}{2}}}}\leq \theta _{n,m}^{(\alpha ,\beta )}\leq {\frac {m\pi }{n+{\tfrac {1}{2}}}}=\theta _{n,m}^{({\frac {1}{2}},-{\frac {1}{2}})}}$
• ${\displaystyle \theta _{n,m}^{(-{\frac {1}{2}},-{\frac {1}{2}})}={\frac {(m-{\tfrac {1}{2}})\pi }{n}}\leq \theta _{n,m}^{(\alpha ,\alpha )}\leq {\frac {m\pi }{n+1}}=\theta _{n,m}^{({\frac {1}{2}},{\frac {1}{2}})}}$ for ${\displaystyle m\leq n/2}$
• ${\displaystyle {{\frac {\left(m+{\tfrac {1}{2}}(\alpha +\beta -1)\right)\pi }{\rho }}<\theta _{n,m}<{\frac {m\pi }{\rho }}}}$ except when ${\displaystyle \alpha ^{2}=\beta ^{2}={\tfrac {1}{4}}}$
• ${\displaystyle \theta _{n,m}^{(\alpha ,\alpha )}>{\frac {\left(m+{\tfrac {1}{2}}\alpha -{\tfrac {1}{4}}\right){\pi }}{n+\alpha +{\tfrac {1}{2}}}}}$ for ${\displaystyle m\leq n/2}$, except when ${\displaystyle \alpha ^{2}={\tfrac {1}{4}}}$
• ${\displaystyle \displaystyle \theta _{n,m}\displaystyle \leq {\frac {j_{\alpha ,m}}{\left(\rho ^{2}+{\tfrac {1}{12}}\left(1-\alpha ^{2}-3\beta ^{2}\right)\right)^{\frac {1}{2}}}}}$
• ${\displaystyle \displaystyle \theta _{n,m}\displaystyle \geq {\frac {j_{\alpha ,m}}{\left(\rho ^{2}+{\tfrac {1}{4}}-{\tfrac {1}{2}}(\alpha ^{2}+\beta ^{2})-{\pi }^{-2}(1-4\alpha ^{2})\right)^{\frac {1}{2}}}}}$ for ${\displaystyle m\leq n/2}$

Fix ${\displaystyle \alpha >-1/2,\beta \geq -1-\alpha }$. Fix ${\displaystyle c\in (0,1)}$.

$${\displaystyle \theta _{n,m}=\phi _{m}+\left(\left(\alpha ^{2}-{\tfrac {1}{4}}\right){\frac {1-\phi _{m}\cot \phi _{m}}{2\phi _{m}}}-{\tfrac {1}{4}}(\alpha ^{2}-\beta ^{2})\tan \left({\tfrac {1}{2}}\phi _{m}\right)\right){\frac {1}{\rho ^{2}}}+\phi _{m}^{2}O\left({\frac {1}{\rho ^{3}}}\right)}$$

uniformly for ${\displaystyle m=1,2,\dots ,\left\lfloor cn\right\rfloor }$.

The zeroes satisfy the Stieltjes relations:^[14][15] $${\displaystyle {\begin{aligned}\sum _{1\leq j\leq n,i\neq j}{\frac {1}{x_{i}-x_{j}}}&={\frac {1}{2}}\left({\frac {\alpha +1}{1-x_{i}}}-{\frac {\beta +1}{1+x_{i}}}\right)\\\sum _{1\leq j\leq n}{\frac {1}{1-x_{j}}}&={\frac {n(n+\alpha +\beta +1)}{2(\alpha +1)}}\\\sum _{1\leq j\leq n}{\frac {1}{1+x_{j}}}&={\frac {n(n+\alpha +\beta +1)}{2(\beta +1)}}\\\sum _{1\leq j\leq n}x_{j}&={\frac {n(\beta -\alpha )}{2n+\alpha +\beta }}\end{aligned}}}$$The first relation can be interpreted physically. Fix an electric particle at +1 with charge ${\displaystyle {\frac {1+\alpha }{2}}}$, and another particle at -1 with charge ${\displaystyle {\frac {1+\beta }{2}}}$. Then, place ${\displaystyle n}$ electric particles with charge ${\displaystyle +1}$. The first relation states that the zeroes of ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ are the equilibrium positions of the particles. This equilibrium is stable and unique.^[15]

Other relations, such as ${\displaystyle \sum _{1\leq j\leq n,i\neq j}{\frac {1}{(x_{i}-x_{j})^{2}}},\sum _{1\leq j\leq n,i\neq j}{\frac {1}{(x_{i}-x_{j})^{3}}}}$, are known in closed form.^[14]

As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Jacobi polynomials.

The electrostatic interpretation allows many relations to be intuitively seen. For example:

• the symmetry relation between ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ and ${\displaystyle P_{n}^{(\beta ,\alpha )}}$;
• the roots monotonically decrease when ${\displaystyle \alpha }$ increases;

Since the Stieltjes relation also exists for the Hermite polynomials and the Laguerre polynomials, by taking an appropriate limit of ${\displaystyle \alpha ,\beta }$, the limit relations are derived. For example, for the Hermite polynomials, the zeros satisfy$${\displaystyle -x_{i}+\sum _{1\leq j\leq n,i\neq j}{\frac {1}{x_{i}-x_{j}}}=0}$$Thus, by taking ${\displaystyle \alpha =\beta \to \infty }$ limit, all the electric particles are forced into an infinitesimal neighborhood of the origin, where the field strength is linear. Then after scaling up the line, we obtain the same electrostatic configuration for the zeroes of Hermite polynomials.

For ${\displaystyle x}$ in the interior of ${\displaystyle [-1,1]}$, the asymptotics of ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ for large ${\displaystyle n}$ is given by the Darboux formula^{[1]: VIII.2}

${\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )=n^{-{\frac {1}{2}}}k(\theta )\cos(N\theta +\gamma )+O\left(n^{-{\frac {3}{2}}}\right),}$

where

${\displaystyle {\begin{aligned}k(\theta )&=\pi ^{-{\frac {1}{2}}}\sin ^{-\alpha -{\frac {1}{2}}}{\tfrac {\theta }{2}}\cos ^{-\beta -{\frac {1}{2}}}{\tfrac {\theta }{2}},\\N&=n+{\tfrac {1}{2}}(\alpha +\beta +1),\\\gamma &=-{\tfrac {\pi }{2}}\left(\alpha +{\tfrac {1}{2}}\right),\\0<\theta &<\pi \end{aligned}}}$

and the "${\displaystyle O}$" term is uniform on the interval ${\displaystyle [\varepsilon ,\pi -\varepsilon ]}$ for every ${\displaystyle \varepsilon >0}$.

For higher orders, define:^[12]

• ${\displaystyle \mathrm {B} }$ is the Euler beta function
• ${\displaystyle (\cdot )_{m}}$ is the falling factorial.
• ${\displaystyle f_{m}(\theta )=\sum _{\ell =0}^{m}{\frac {C_{m,\ell }(\alpha ,\beta )}{\ell !(m-\ell )!}}{\frac {\cos \theta _{n,m,\ell }}{\left(\sin {\frac {1}{2}}\theta \right)^{\ell }\left(\cos {\frac {1}{2}}\theta \right)^{m-\ell }}}}$
• ${\displaystyle C_{m,\ell }(\alpha ,\beta )={\left({\tfrac {1}{2}}+\alpha \right)_{\ell }}{\left({\tfrac {1}{2}}-\alpha \right)_{\ell }}{\left({\tfrac {1}{2}}+\beta \right)_{m-\ell }}{\left({\tfrac {1}{2}}-\beta \right)_{m-\ell }}}$
• ${\displaystyle \theta _{n,m,\ell }={\tfrac {1}{2}}(2n+\alpha +\beta +m+1)\theta -{\tfrac {1}{2}}(\alpha +\ell +{\tfrac {1}{2}})\pi }$

Fix real ${\displaystyle \alpha ,\beta }$, fix ${\displaystyle M=1,2,\dots }$, fix ${\displaystyle \delta \in (0,\pi /2)}$. As ${\displaystyle n\to \infty }$,$${\displaystyle \left(\sin {\tfrac {1}{2}}\theta \right)^{\alpha +{\frac {1}{2}}}\left(\cos {\tfrac {1}{2}}\theta \right)^{\beta +{\frac {1}{2}}}P_{n}^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-1}2^{2n+\alpha +\beta +1}\mathrm {B} \left(n+\alpha +1,n+\beta +1\right)\left(\sum _{m=0}^{M-1}{\frac {f_{m}(\theta )}{2^{m}{\left(2n+\alpha +\beta +2\right)_{m}}}}+O\left(n^{-M}\right)\right)}$$uniformly for all ${\displaystyle \theta \in [\delta ,\pi -\delta ]}$.

The ${\displaystyle M=1}$ case is the above Darboux formula.

Define:^[12]

• ${\displaystyle J_{\nu }}$ is the Bessel function
• ${\displaystyle \rho =n+{\tfrac {1}{2}}(\alpha +\beta +1)}$
• ${\displaystyle g(\theta )=\left({\tfrac {1}{4}}-\alpha ^{2}\right)\left(\cot \left({\tfrac {1}{2}}\theta \right)-\left({\tfrac {1}{2}}\theta \right)^{-1}\right)-\left({\tfrac {1}{4}}-\beta ^{2}\right)\tan \left({\tfrac {1}{2}}\theta \right)}$

Fix real ${\displaystyle \alpha ,\beta }$, fix ${\displaystyle M=0,1,2,\dots }$. As ${\displaystyle n\to \infty }$, we have the Hilb's type formula:^[16]$${\displaystyle (\sin {\tfrac {1}{2}}\theta )^{\alpha +{\frac {1}{2}}}(\cos {\tfrac {1}{2}}\theta )^{\beta +{\frac {1}{2}}}P_{n}^{(\alpha ,\beta )}\left(\cos \theta \right)={\frac {\Gamma \left(n+\alpha +1\right)}{2^{\frac {1}{2}}\rho ^{\alpha }n!}}\left(\theta ^{\frac {1}{2}}J_{\alpha }\left(\rho \theta \right)\sum _{m=0}^{M}{\dfrac {A_{m}(\theta )}{\rho ^{2m}}}+\theta ^{\frac {3}{2}}J_{\alpha +1}\left(\rho \theta \right)\sum _{m=0}^{M-1}{\dfrac {B_{m}(\theta )}{\rho ^{2m+1}}}+\varepsilon _{M}(\rho ,\theta )\right)}$$where ${\displaystyle A_{m},B_{m}}$ are functions of ${\displaystyle \theta }$. The first few entries are:$${\displaystyle {\begin{aligned}A_{0}(\theta )&=1\\\theta B_{0}(\theta )&={\frac {1}{4}}g(\theta )\\A_{1}(\theta )&={\frac {1}{8}}g^{\prime }(\theta )-{\frac {1+2\alpha }{8}}{\frac {g(\theta )}{\theta }}-{\frac {1}{32}}(g(\theta ))^{2}\end{aligned}}}$$

For any fixed arbitrary constant ${\displaystyle c>0}$, the error term satisfies$${\displaystyle \varepsilon _{M}(\rho ,\theta )={\begin{cases}\theta O\left(\rho ^{-2M-(3/2)}\right),&c\rho ^{-1}\leq \theta \leq \pi -\delta ,\\\theta ^{\alpha +(5/2)}O\left(\rho ^{-2M+\alpha }\right),&0\leq \theta \leq c\rho ^{-1},\end{cases}}}$$

The asymptotics of the Jacobi polynomials near the points ${\displaystyle \pm 1}$ is given by the Mehler–Heine formula

${\displaystyle {\begin{aligned}\lim _{n\to \infty }n^{-\alpha }P_{n}^{(\alpha ,\beta )}\left(\cos \left({\tfrac {z}{n}}\right)\right)&=\left({\tfrac {z}{2}}\right)^{-\alpha }J_{\alpha }(z)\\\lim _{n\to \infty }n^{-\beta }P_{n}^{(\alpha ,\beta )}\left(\cos \left(\pi -{\tfrac {z}{n}}\right)\right)&=\left({\tfrac {z}{2}}\right)^{-\beta }J_{\beta }(z)\end{aligned}}}$

where the limits are uniform for ${\displaystyle z}$ in a bounded domain.

The asymptotics outside ${\displaystyle [-1,1]}$ is less explicit.

The expression (1) allows the expression of the Wigner d-matrix ${\displaystyle d_{m',m}^{j}(\phi )}$ (for ${\displaystyle 0\leq \phi \leq 4\pi }$) in terms of Jacobi polynomials:^[17]

${\displaystyle d_{m'm}^{j}(\phi )=(-1)^{\frac {m-m'-|m-m'|}{2}}\left[{\frac {(j+M)!(j-M)!}{(j+N)!(j-N)!}}\right]^{\frac {1}{2}}\left(\sin {\tfrac {\phi }{2}}\right)^{|m-m'|}\left(\cos {\tfrac {\phi }{2}}\right)^{|m+m'|}P_{j-M}^{(|m-m'|,|m+m'|)}(\cos \phi ),}$

where ${\displaystyle M=\max(|m|,|m'|),N=\min(|m|,|m'|)}$.

• Askey–Gasper inequality
• Big q-Jacobi polynomials
• Continuous q-Jacobi polynomials
• Little q-Jacobi polynomials
• Pseudo Jacobi polynomials
• Gegenbauer polynomials
• Romanovski polynomials