Gold 56

Anger function

Gold ID: 56
Link: https://sigir21.wmflabs.org/wiki/Anger_function#math.108.3
Formula: $J_{ν} (z) = \cos \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k}}{4^{k} Γ (k + \frac{ν}{2} + 1) Γ (k - \frac{ν}{2} + 1)} + \sin \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k + 1}}{2^{2 k + 1} Γ (k + \frac{ν}{2} + \frac{3}{2}) Γ (k - \frac{ν}{2} + \frac{3}{2})}$
TeX Source: \mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Translation: \AngerJ{\nu}@{z} = \cos \frac{\cpi \nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{4^k \Gamma(k + \frac{\nu}{2} + 1) \Gamma(k - \frac{\nu}{2} + 1)} + \sin \frac{\cpi \nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{2^{2k+1} \Gamma(k + \frac{\nu}{2} + \frac{3}{2}) \Gamma(k - \frac{\nu}{2} + \frac{3}{2})}
Expected (Gold Entry): \AngerJ{\nu}@{z} = \cos \frac{\cpi\nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{4^k\EulerGamma@{k+\frac{\nu}{2}+1}\EulerGamma@{k-\frac{\nu}{2}+1}}+\sin\frac{\cpi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^k z^{2k+1}}{2^{2k+1}\EulerGamma@{k+\frac{\nu}{2}+\frac{3}{2}}\EulerGamma@{k-\frac{\nu}{2}+\frac{3}{2}}}

Translation: AngerJ[\[Nu], z] == Cos[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \[CapitalGamma]*(k +Divide[\[Nu],2]+ 1)*\[CapitalGamma]*(k -Divide[\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \[CapitalGamma]*(k +Divide[\[Nu],2]+Divide[3,2])*\[CapitalGamma]*(k -Divide[\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]
Expected (Gold Entry): AngerJ[\[Nu], z] == Cos[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* Gamma[k +Divide[\[Nu],2]+ 1]*Gamma[k -Divide[\[Nu],2]+ 1]], {k, 0, Infinity}]+ Sin[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* Gamma[k +Divide[\[Nu],2]+Divide[3,2]]*Gamma[k -Divide[\[Nu],2]+Divide[3,2]]], {k, 0, Infinity}]

Translation: AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* Gamma*(k +(nu)/(2)+ 1)*Gamma*(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* Gamma*(k +(nu)/(2)+(3)/(2))*Gamma*(k -(nu)/(2)+(3)/(2))), k = 0..infinity)
Expected (Gold Entry): AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* GAMMA(k +(nu)/(2)+ 1)*GAMMA(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* GAMMA(k +(nu)/(2)+(3)/(2))*GAMMA(k -(nu)/(2)+(3)/(2))), k = 0..infinity)