Coulomb Functions - 33.6 Power-Series Expansions in

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
33.6.E3	${\displaystyle{\displaystyle(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell}}}$ (k+\ell)(k-\ell-1)A_{k}^{\ell} = 2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell}	${\displaystyle{\displaystyle k=\ell+3}}$	(k + ell)*(k - ell - 1)*(A[k])^(ell) = 2*eta*(A[k - 1])^(ell)- (A[k - 2])^(ell)	(k + \[ScriptL])*(k - \[ScriptL]- 1)*(Subscript[A, k])^\[ScriptL] == 2*\[Eta]*(Subscript[A, k - 1])^\[ScriptL]- (Subscript[A, k - 2])^\[ScriptL]	Skipped - no semantic math	Skipped - no semantic math	-	-
33.6.E4	${\displaystyle{\displaystyle A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}}\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right)}}$ A_{k}^{\ell}(\eta) = \dfrac{(-\iunit)^{k-\ell-1}}{(k-\ell-1)!}\*\genhyperF{2}{1}@{\ell+1-k,\ell+1-\iunit\eta}{2\ell+2}{2}		(A[k])^(ell)(eta) = ((- I)^(k - ell - 1))/(factorial(k - ell - 1))* hypergeom([ell + 1 - k , ell + 1 - I*eta], [2*ell + 2], 2)	(Subscript[A, k])^\[ScriptL][\[Eta]] == Divide[(- I)^(k - \[ScriptL]- 1),(k - \[ScriptL]- 1)!]* HypergeometricPFQ[{\[ScriptL]+ 1 - k , \[ScriptL]+ 1 - I*\[Eta]}, {2*\[ScriptL]+ 2}, 2]	Failure	Failure	Error	Failed [293 / 300] Result: Complex[0.5000000000000001, 0.8660254037844386] Test Values: {Rule[k, 1], Rule[ℓ, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, 1.0] Test Values: {Rule[k, 1], Rule[ℓ, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data