Orthogonal Polynomials - 18.35 Pollaczek Polynomials

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.35.E4	${\displaystyle{\displaystyle\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n}}}{n!}e^{\mathrm{i}n\theta}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm{i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}(\theta)};e^{-2\mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)_{\ell}}}{\ell!}\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n-\ell}}}{(n-\ell)!}e^{\mathrm{i}(n-2\ell)\theta}}}$ \frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi}}$	(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n))/(factorial(n))*exp(I*n*theta)* hypergeom([- n , lambda + I*((a*cos(theta)+ b)/(sin(theta)))], [- n - lambda + 1 + I*((a*cos(theta)+ b)/(sin(theta)))], exp(- 2*I*theta)) = sum((pochhammer(lambda + I*((a*cos(theta)+ b)/(sin(theta))), ell))/(factorial(ell))*(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n - ell))/(factorial(n - ell))*exp(I*(n - 2*ell)*theta), ell = 0..n)	Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n],(n)!]*Exp[I*n*\[Theta]]* HypergeometricPFQ[{- n , \[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, {- n - \[Lambda]+ 1 + I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, Exp[- 2*I*\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), \[ScriptL]],(\[ScriptL])!]*Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n - \[ScriptL]],(n - \[ScriptL])!]*Exp[I*(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]	Error	Successful	-	Successful [Tested: 300]
18.35.E6	${\displaystyle{\displaystyle w^{(\lambda)}(\cos\theta;a,b)=\pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin\theta)^{2\lambda-1}\*{\left|\Gamma\left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)\right|^{2}}}}$ w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}	${\displaystyle{\displaystyle a\geq b,b\geq-a,\lambda>-\frac{1}{2},0<\theta,\theta<\pi}}$	(w(cos(theta); a , b))^(lambda) = (Pi)^(- 1)* (2)^(2*lambda - 1)* exp((2*theta - Pi)*((a*cos(theta)+ b)/(sin(theta))))*(sin(theta))^(2*lambda - 1)* (abs(GAMMA(lambda + I*((a*cos(theta)+ b)/(sin(theta))))))^(2)	(w[Cos[\[Theta]]; a , b])^(\[Lambda]) == (Pi)^(- 1)* (2)^(2*\[Lambda]- 1)* Exp[(2*\[Theta]- Pi)*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]*(Sin[\[Theta]])^(2*\[Lambda]- 1)* (Abs[Gamma[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]])^(2)	Translation Error	Translation Error	-	-