Legendre and Related Functions - 14.28 Sums

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.28.E1	${\displaystyle{\displaystyle P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right)}}$ \assLegendreP[]{\nu}@{z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos@@{\phi}} = \assLegendreP[]{\nu}@{z_{1}}\assLegendreP[]{\nu}@{z_{2}}+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\*\assLegendreP[m]{\nu}@{z_{1}}\assLegendreP[m]{\nu}(z_{2})\cos@{m\phi}	${\displaystyle{\displaystyle\Re(\nu-m+1)>0,\Re(\nu+m+1)>0}}$	LegendreP(nu, z[1]*z[2]-((z[1])^(2)- 1)^(1/2)*((z[2])^(2)- 1)^(1/2)* cos(phi)) = LegendreP(nu, z[1])*LegendreP(nu, z[2])+ 2*sum((- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))* LegendreP(nu, m, z[1])*LegendreP(nu, m, z[2])*cos(m*phi), m = 1..infinity)	LegendreP[\[Nu], 0, 3, Subscript[z, 1]*Subscript[z, 2]-((Subscript[z, 1])^(2)- 1)^(1/2)*((Subscript[z, 2])^(2)- 1)^(1/2)* Cos[\[Phi]]] == LegendreP[\[Nu], 0, 3, Subscript[z, 1]]*LegendreP[\[Nu], 0, 3, Subscript[z, 2]]+ 2*Sum[(- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]* LegendreP[\[Nu], m, 3, Subscript[z, 1]]*LegendreP[\[Nu], m, 3, Subscript[z, 2]]*Cos[m*\[Phi]], {m, 1, Infinity}, GenerateConditions->None]	Translation Error	Translation Error	-	-
14.28.E2	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_{n}\left(z_{2}\right)=\frac{1}{z_{1}-z_{2}}}}$ \sum_{n=0}^{\infty}(2n+1)\assLegendreQ[]{n}@{z_{1}}\assLegendreP[]{n}@{z_{2}} = \frac{1}{z_{1}-z_{2}}		sum((2*n + 1)*LegendreQ(n, z[1])*LegendreP(n, z[2]), n = 0..infinity) = (1)/(z[1]- z[2])	Sum[(2*n + 1)*LegendreQ[n, 0, 3, Subscript[z, 1]]*LegendreP[n, 0, 3, Subscript[z, 2]], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Subscript[z, 1]- Subscript[z, 2]]	Failure	Failure	Skipped - Because timed out	Failed [100 / 100] Result: Plus[DirectedInfinity[], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.6830127018922194, -0.18301270189221946], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], LegendreQ[n, 0, 3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data