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DLMF Formula Constraints Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
1.2.E1 ( n k ) = n ! ( n - k ) ! ⁒ k ! binomial 𝑛 π‘˜ 𝑛 𝑛 π‘˜ π‘˜ {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}=\frac{n!}{(n-k)!k!}}}
\binom{n}{k} = \frac{n!}{(n-k)!k!}		 

binomial(n,k) = (factorial(n))/(factorial(n - k)*factorial(k))

Binomial[n,k] == Divide[(n)!,(n - k)!*(k)!]
Successful Successful - Successful [Tested: 9]
18.35.E4 ( Ξ» - i ⁒ Ο„ a , b ⁒ ( ΞΈ ) ) n n ! ⁒ e i ⁒ n ⁒ ΞΈ ⁒ F 1 2 ⁑ ( - n , Ξ» + i ⁒ Ο„ a , b ⁒ ( ΞΈ ) - n - Ξ» + 1 + i ⁒ Ο„ a , b ⁒ ( ΞΈ ) ; e - 2 ⁒ i ⁒ ΞΈ ) = βˆ‘ β„“ = 0 n ( Ξ» + i ⁒ Ο„ a , b ⁒ ( ΞΈ ) ) β„“ β„“ ! ⁒ ( Ξ» - i ⁒ Ο„ a , b ⁒ ( ΞΈ ) ) n - β„“ ( n - β„“ ) ! ⁒ e i ⁒ ( n - 2 ⁒ β„“ ) ⁒ ΞΈ Pochhammer πœ† imaginary-unit subscript 𝜏 π‘Ž 𝑏 πœƒ 𝑛 𝑛 superscript 𝑒 imaginary-unit 𝑛 πœƒ Gauss-hypergeometric-F-as-2F1 𝑛 πœ† imaginary-unit subscript 𝜏 π‘Ž 𝑏 πœƒ 𝑛 πœ† 1 imaginary-unit subscript 𝜏 π‘Ž 𝑏 πœƒ superscript 𝑒 2 imaginary-unit πœƒ superscript subscript β„“ 0 𝑛 Pochhammer πœ† imaginary-unit subscript 𝜏 π‘Ž 𝑏 πœƒ β„“ β„“ Pochhammer πœ† imaginary-unit subscript 𝜏 π‘Ž 𝑏 πœƒ 𝑛 β„“ 𝑛 β„“ superscript 𝑒 imaginary-unit 𝑛 2 β„“ πœƒ {\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}		 0 < ΞΈ , ΞΈ < Ο€ formulae-sequence 0 πœƒ πœƒ πœ‹ {\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]
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