Jacobi: Special cases

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Jacobi: Special cases

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Koornwinder Addendum: Jacobi

Orthogonality relation

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}(P^{(\alpha,\beta)}_{n}\left(1\right))^{2}}=\frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1}\frac{{\left(\beta+1\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\alpha+\beta+2\right)_{n}}}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1}\frac{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}n!}h_{0}=\frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1}\frac{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}n!}h_{0}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{2^{\alpha+\beta+1}\Gamma\left(\alpha+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}=\frac{2^{\alpha+\beta+1}\Gamma\left(\alpha+1\right)\Gamma\left(-\alpha-\beta-1\right)}{\Gamma\left(-\beta\right)}}}}$

${\displaystyle{\displaystyle{\displaystyle\int_{1}^{\infty}P^{(\alpha,\beta)}_{m}\left(x\right)P^{(\alpha,\beta)}_{n}\left(x\right)(x-1)^{\alpha}(x+1)^{\beta}dx=h_{n}\delta_{m,n}}}}$
${\displaystyle{\displaystyle{\displaystyle-1-\beta>\alpha>-1,m,n<-\frac{1}{2}(\alpha+\beta+1)}}}$

Symmetry

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(-x\right)=(-1)^{n}P^{(\beta,\alpha)}_{n}\left(x\right)}}}$

Jacobi: Special cases: Special values

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}P^{(\alpha,\beta)}_{n}\left(-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{n!}\frac{P^{(\alpha,\beta)}_{n}\left(-1\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}}}}}$

Bilateral generating functions

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}r^{n}P^{(\alpha,\beta)}_{n}\left(x\right)P^{(\alpha,\beta)}_{n}\left(y\right)=\frac{1}{(1+r)^{\alpha+\beta+1}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+1)}{\frac{1}{2}(\alpha+\beta+2)}{\alpha+1}{\beta+1}{\frac{r(1-x)(1-y)}{(1+r)^{2}}}{\frac{r(1+x)(1+y)}{(1+r)^{2}}}}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{2n+\alpha+\beta+1}{n+\alpha+\beta+1}\frac{{\left(\alpha+\beta+2\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}r^{n}P^{(\alpha,\beta)}_{n}\left(x\right)P^{(\alpha,\beta)}_{n}\left(y\right)=\frac{1-r}{(1+r)^{\alpha+\beta+2}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+2)}{\frac{1}{2}(\alpha+\beta+3)}{\alpha+1}{\beta+1}{\frac{r(1-x)(1-y)}{(1+r)^{2}}}{\frac{r(1+x)(1+y)}{(1+r)^{2}}}}}}$

Quadratic transformations

${\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n}\left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n}\left(1\right)}=\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n}\left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n}\left(1\right)}=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,-\frac{1}{2})}_{n}\left(1\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n+1}\left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n+1}\left(1\right)}=\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}\left(1\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n+1}\left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1\right)}}}}$

Differentiation formulas

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left((1-x)^{\alpha}P^{(\alpha,\beta)}_{n}\left(x\right)\right)=-(n+\alpha)(1-x)^{\alpha-1}P^{(\alpha-1,\beta+1)}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\left((1-x)\frac{d}{dx}-\alpha\right)P^{(\alpha,\beta)}_{n}\left(x\right)=-(n+\alpha)P^{(\alpha-1,\beta+1)}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left((1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\right)=(n+\beta)(1+x)^{\beta-1}P^{(\alpha+1,\beta-1)}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\left((1+x)\frac{d}{dx}+\beta\right)P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\beta)P^{(\alpha+1,\beta-1)}_{n}\left(x\right)}}}$

Generalized Gegenbauer polynomials

${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right):=\mathrm{c}onstP^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right),\qquad S^{(\alpha,\beta)}_{2m+1}\left(x\right):}}}$
${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right)=\mathrm{c}onstxP^{(\alpha,\beta+1)}_{m}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}S^{(\alpha,\beta)}_{m}\left(x\right)S^{(\alpha,\beta)}_{n}\left(x\right)|x|^{2\beta+1}(1-x^{2})^{\alpha}dx=0\qquad(m\neq n)}}}$
${\displaystyle{\displaystyle{\displaystyle(T_{\mu}f)(x):=f^{\prime}(x)+\mu\frac{f(x)-f(-x)}{x}}}}$
${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right)=\frac{{\left(\alpha+\beta+1\right)_{m}}}{{\left(\beta+1\right)_{m}}}P^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right)S^{(\alpha,\beta)}_{2m+1}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right)=\frac{{\left(\alpha+\beta+1\right)_{m+1}}}{{\left(\beta+1\right)_{m+1}}}xP^{(\alpha,\beta+1)}_{m}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle T_{\beta+\frac{1}{2}}S^{(\alpha,\beta)}_{n}=2(\alpha+\beta+1)S^{(\alpha+1,\beta)}_{n-1}}}}$
${\displaystyle{\displaystyle{\displaystyle T_{\beta+\frac{1}{2}}^{2}S^{(\alpha,\beta)}_{n}=4(\alpha+\beta+1)(\alpha+\beta+2)S^{(\alpha+2,\beta)}_{n-2}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(\frac{d^{2}}{dx^{2}}+\frac{2\beta+1}{x}\frac{d}{dx}\right)P^{(\alpha,\beta)}_{n}\left(2x^{2}-1\right)=4(n+\alpha+\beta+1)(n+\beta)P^{(\alpha+2,\beta)}_{n-1}\left(2x^{2}-1\right)}}}$

Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\lambda\neq 0}}}$

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{\lambda}_{m}\left(x\right)C^{\lambda}_{n}\left(x\right)\,dx{}=\frac{\pi\Gamma\left(n+2\lambda\right)2^{1-2\lambda}}{\left\{\Gamma\left(\lambda\right)\right\}^{2}(n+\lambda)n!}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\lambda>-\frac{1}{2}\quad\lambda\neq 0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2(n+\lambda)xC^{\lambda}_{n}\left(x\right)=(n+1)C^{\lambda}_{n+1}\left(x\right)+(n+2\lambda-1)C^{\lambda}_{n-1}\left(x\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}^{\lambda}_{n}\left(x\right){x}={\widehat{C}}^{\lambda}_{n+1}\left(x\right){x}+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}{\widehat{C}}^{\lambda}_{n-1}\left(x\right){x}}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac{2^{n}{\left(\lambda\right)_{n}}}{n!}{\widehat{C}}^{\lambda}_{n}\left(x\right){x}}}}$

Differential equation

${\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)-(2\lambda+1)xy^{\prime}(x)+n(n+2\lambda)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=C^{\lambda}_{n}\left(x\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}C^{\lambda}_{n}\left(x\right)=2\lambda C^{\lambda+1}_{n-1}\left(x\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(1-x^{2})\frac{d}{dx}C^{\lambda}_{n}\left(x\right)+(1-2\lambda)xC^{\lambda}_{n}\left(x\right)=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}C^{\lambda-1}_{n+1}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[(1-x^{2})^{\lambda-\textstyle\frac{1}{2}}C^{\lambda}_{n}\left(x\right)\right]{}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^{2})^{\lambda-\textstyle\frac{3}{2}}C^{\lambda-1}_{n+1}\left(x\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{\lambda-\frac{1}{2}}C^{\lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}(-1)^{n}}{{\left(\lambda+\frac{1}{2}\right)_{n}}2^{n}n!}\left(\frac{d}{dx}\right)^{n}\left[(1-x^{2})^{\lambda+n-\frac{1}{2}}\right]}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle(1-2xt+t^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{\lambda}_{n}\left(x\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty}\frac{{\left(\lambda+\frac{1}{2}\right)_{n}}}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}$

${\displaystyle{\displaystyle{\displaystyle\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}}=\sum_{n=0}^{\infty}\frac{C^{\lambda}_{n}\left(x\right)}{{\left(2\lambda\right)_{n}}{\left(\lambda+\frac{1}{2}\right)_{n}}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{C^{\lambda}_{n}\left(x\right)}{{\left(2\lambda\right)_{n}}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(2\lambda-\gamma\right)_{n}}}{{\left(2\lambda\right)_{n}}{\left(\lambda+\frac{1}{2}\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

${\displaystyle{\displaystyle{\displaystyle(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x^{2}-1)t^{2}}{(1-xt)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relation

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha^{-\frac{1}{2}n}C^{\alpha+\frac{1}{2}}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n}\left(x\right)=\frac{{\left(\lambda\right)_{n}}}{{\left(\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n+1}\left(x\right)=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}}$

Koornwinder Addendum: Gegenbauer

Orthogonality relation

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\lambda}{\lambda+n}\frac{{\left(2\lambda\right)_{n}}}{n!}h_{0}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{{\pi^{\frac{missing}{missing}}}12\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma\left(\lambda+1\right)}\frac{h_{n}}{h_{0}(C^{\lambda}_{n}\left(1\right))^{2}}=\frac{\lambda}{\lambda+n}\frac{n!}{{\left(2\lambda\right)_{n}}}}}}$

Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}\HyperpFq{2}{1}@@{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}}}}$

Jacobi: Special cases: Special value

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(1\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}}}}$

Expression in terms of Jacobi

${\displaystyle{\displaystyle{\displaystyle\frac{C^{\lambda}_{n}\left(x\right)}{C^{\lambda}_{n}\left(1\right)}=\frac{P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}{P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(1\right)},\qquad C^{\lambda}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{C^{\lambda}_{n}\left(x\right)}{C^{\lambda}_{n}\left(1\right)}=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}}$

Re: (9.8.21)

${\displaystyle{\displaystyle{\displaystyle x^{2}C^{\lambda}_{n}\left(x\right)=\frac{(n+1)(n+2)}{4(n+\lambda)(n+\lambda+1)}C^{\lambda}_{n+2}\left(x\right)+\frac{n^{2}+2n\lambda+\lambda-1}{2(n+\lambda-1)(n+\lambda+1)}C^{\lambda}_{n}\left(x\right)+\frac{(n+2\lambda-1)(n+2\lambda-2)}{4(n+\lambda)(n+\lambda-1)}C^{\lambda}_{n-2}\left(x\right)}}}$

Bilateral generating functions

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{n!}{{\left(2\lambda\right)_{n}}}r^{n}C^{\lambda}_{n}\left(x\right)C^{\lambda}_{n}\left(y\right)=\frac{1}{(1-2rxy+r^{2})^{\lambda}}\HyperpFq{2}{1}@@{\frac{1}{2}\lambda,\frac{1}{2}(\lambda+1)}{\lambda+\frac{1}{2}}{\frac{4r^{2}(1-x^{2})(1-y^{2})}{(1-2rxy+r^{2})^{2}}}(r\in(-1,1),\;x,y\in[-1,1])}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\lambda+n}{\lambda}\frac{n!}{{\left(2\lambda\right)_{n}}}r^{n}C^{\lambda}_{n}\left(x\right)C^{\lambda}_{n}\left(y\right)=\frac{1-r^{2}}{(1-2rxy+r^{2})^{\lambda+1}}\HyperpFq{2}{1}@@{\frac{1}{2}(\lambda+1),\frac{1}{2}(\lambda+2)}{\lambda+\frac{1}{2}}{\frac{4r^{2}(1-x^{2})(1-y^{2})}{(1-2rxy+r^{2})^{2}}}(r\in(-1,1),\;x,y\in[-1,1])}}}$

Trigonometric expansions

${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)=\sum_{k=0}^{n}\frac{{\left(\lambda\right)_{k}}{\left(\lambda\right)_{n-k}}}{k!(n-k)!}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)={\mathrm{e}^{\mathrm{i}n\theta}}\frac{{\left(\lambda\right)_{n}}}{n!}\HyperpFq{2}{1}@@{-n,\lambda}{1-\lambda-n}{{\mathrm{e}^{-2\mathrm{i}\theta}}}}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)=\frac{{\left(\lambda\right)_{n}}}{2^{\lambda}n!}{\mathrm{e}^{-\frac{1}{2}\mathrm{i}\lambda\pi}}{\mathrm{e}^{\mathrm{i}(n+\lambda)\theta}}(\sin\theta)^{-\lambda}\HyperpFq{2}{1}@@{\lambda,1-\lambda}{1-\lambda-n}{\frac{\mathrm{i}{\mathrm{e}^{-\mathrm{i}\theta}}}{2\sin\theta}}}}}$
${\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta\right)=\frac{{\left(\lambda\right)_{n}}}{n!}\sum_{k=0}^{\infty}\frac{{\left(\lambda\right)_{k}}{\left(1-\lambda\right)_{k}}}{{\left(1-\lambda-n\right)_{k}}k!}\frac{\cos\left((n-k+\lambda)\theta+\frac{1}{2}(k-\lambda)\pi\right)}{(2\sin\theta)^{k+\lambda}}}}}$
$C_n^{\mathrm{λ}}⁡\left(\cos ⁡\mathrm{θ}\right)=\frac{2⁢\mathrm{Γ}⁡\left(\mathrm{λ}+\frac{1}{2}\right)}{{\mathrm{π}}^{\frac{\mathrm{missing}}{\mathrm{missing}}}⁢12⁢\mathrm{Γ}⁡\left(\mathrm{λ}+1\right)}⁢\frac{{\left(2⁢\mathrm{λ}\right)}_n}{{\left(\mathrm{λ}+1\right)}_n}⁢{(\sin ⁡\mathrm{θ})}^{1-2⁢\mathrm{λ}}⁢\underset{k=0}{\overset{\mathrm{∞}}{∑}}\frac{{\left(1-\mathrm{λ}\right)}_k⁢{\left(n+1\right)}_k}{{\left(n+\mathrm{λ}+1\right)}_k⁢k!}⁢\sin ⁡\left((2⁢k+n+1)⁢\mathrm{θ}\right)$