3.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
3.9.E1	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{t_{n}-\sigma}{s_{n}-\sigma}=0}}$ \lim_{n\to\infty}\frac{t_{n}-\sigma}{s_{n}-\sigma} = 0		limit((t[n]- sigma)/(s[n]- sigma), n = infinity) = 0	Limit[Divide[Subscript[t, n]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E7	${\displaystyle{\displaystyle t_{n}=s_{n}-\frac{(\Delta s_{n})^{2}}{\Delta^{2}s_{n}}}}$ t_{n} = s_{n}-\frac{(\Delta s_{n})^{2}}{\Delta^{2}s_{n}}		t[n] = s[n]-((Delta*s[n])^(2))/((Delta)^(2)* s[n])	Subscript[t, n] == Subscript[s, n]-Divide[(\[CapitalDelta]*Subscript[s, n])^(2),\[CapitalDelta]^(2)* Subscript[s, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E8	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma}=\rho}}$ \lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = \rho	${\displaystyle{\displaystyle\left|\rho\right|<1}}$	limit((s[n + 1]- sigma)/(s[n]- sigma), n = infinity) = rho	Limit[Divide[Subscript[s, n + 1]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == \[Rho]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E9	${\displaystyle{\displaystyle t_{n,2k}=\frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})}}}$ t_{n,2k} = \frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})}		t[n , 2*k] = (H[k + 1](s[n]))/(H[k]((Delta)^(2)* s[n]))	Subscript[t, n , 2*k] == Divide[Subscript[H, k + 1][Subscript[s, n]],Subscript[H, k][\[CapitalDelta]^(2)* Subscript[s, n]]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9#Ex1	${\displaystyle{\displaystyle\varepsilon_{-1}^{(n)}=0}}$ \varepsilon_{-1}^{(n)} = 0		(varepsilon[- 1])^(n) = 0	(Subscript[\[CurlyEpsilon], - 1])^(n) == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9#Ex2	${\displaystyle{\displaystyle\varepsilon_{0}^{(n)}=s_{n}}}$ \varepsilon_{0}^{(n)} = s_{n}		(varepsilon[0])^(n) = s[n]	(Subscript[\[CurlyEpsilon], 0])^(n) == Subscript[s, n]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9#Ex3	${\displaystyle{\displaystyle\varepsilon_{m+1}^{(n)}=\varepsilon_{m-1}^{(n+1)}+\frac{1}{\varepsilon_{m}^{(n+1)}-\varepsilon_{m}^{(n)}}}}$ \varepsilon_{m+1}^{(n)} = \varepsilon_{m-1}^{(n+1)}+\frac{1}{\varepsilon_{m}^{(n+1)}-\varepsilon_{m}^{(n)}}		(varepsilon[m + 1])^(n) = (varepsilon[m - 1])^(n + 1)+(1)/((varepsilon[m])^(n + 1)- (varepsilon[m])^(n))	(Subscript[\[CurlyEpsilon], m + 1])^(n) == (Subscript[\[CurlyEpsilon], m - 1])^(n + 1)+Divide[1,(Subscript[\[CurlyEpsilon], m])^(n + 1)- (Subscript[\[CurlyEpsilon], m])^(n)]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E12	${\displaystyle{\displaystyle s_{n}=\sum_{j=1}^{n}\frac{(-1)^{j+1}}{j^{2}}}}$ s_{n} = \sum_{j=1}^{n}\frac{(-1)^{j+1}}{j^{2}}		s[n] = sum(((- 1)^(j + 1))/((j)^(2)), j = 1..n)	Subscript[s, n] == Sum[Divide[(- 1)^(j + 1),(j)^(2)], {j, 1, n}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E13	${\displaystyle{\displaystyle{\cal L}_{k}^{(n)}(s)=\frac{\sum_{j=0}^{k}(-1)^{j}\genfrac{(}{)}{0.0pt}{}{k}{j}c_{j,k,n}\ifrac{s_{n+j}}{a_{n+j+1}}}{\sum_{j=0}^{k}(-1)^{j}\genfrac{(}{)}{0.0pt}{}{k}{j}c_{j,k,n}/a_{n+j+1}}}}$ {\cal L}_{k}^{(n)}(s) = \frac{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}\ifrac{s_{n+j}}{a_{n+j+1}}}{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}/a_{n+j+1}}		(L[k])^(n)(s) = (sum((- 1)^(j)*binomial(k,j)*(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1)))*(s[n + j])/(a[n + j + 1]), j = 0..k))/(sum((- 1)^(j)*binomial(k,j)*(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1)))/a[n + j + 1], j = 0..k))	(Subscript[L, k])^(n)[s] == Divide[Sum[(- 1)^(j)*Binomial[k,j]*(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)])*Divide[Subscript[s, n + j],Subscript[a, n + j + 1]], {j, 0, k}, GenerateConditions->None],Sum[(- 1)^(j)*Binomial[k,j]*(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)])/Subscript[a, n + j + 1], {j, 0, k}, GenerateConditions->None]]	Failure	Failure	Error	Skipped - Because timed out
3.9.E15	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma}=1}}$ \lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = 1		limit((s[n + 1]- sigma)/(s[n]- sigma), n = infinity) = 1	Limit[Divide[Subscript[s, n + 1]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
3.9.E16	${\displaystyle{\displaystyle c_{j,k,n}=\frac{{\left(\beta+n+j\right)_{k-1}}}{{\left(\beta+n+k\right)_{k-1}}}}}$ c_{j,k,n} = \frac{\Pochhammersym{\beta+n+j}{k-1}}{\Pochhammersym{\beta+n+k}{k-1}}		(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1))) = (pochhammer(beta + n + j, k - 1))/(pochhammer(beta + n + k, k - 1))	(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)]) == Divide[Pochhammer[\[Beta]+ n + j, k - 1],Pochhammer[\[Beta]+ n + k, k - 1]]	Failure	Failure	Failed [36 / 81] Result: -.277777778e-1 Test Values: {beta = 1.5, j = 1, k = 2, n = 1} Result: -.181818182e-1 Test Values: {beta = 1.5, j = 1, k = 2, n = 2} Result: -.128205129e-1 Test Values: {beta = 1.5, j = 1, k = 2, n = 3} Result: -.805594406e-1 Test Values: {beta = 1.5, j = 1, k = 3, n = 1} ... skip entries to safe data	Failed [36 / 81] Result: -0.02777777777777768 Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 1], Rule[β, 1.5]} Result: -0.018181818181818188 Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 2], Rule[β, 1.5]} ... skip entries to safe data
3.9.E17	${\displaystyle{\displaystyle c_{j,k,n}=\frac{{\left(-\gamma-n-j\right)_{k-1}}}{{\left(-\gamma-n-k\right)_{k-1}}}}}$ c_{j,k,n} = \frac{\Pochhammersym{-\gamma-n-j}{k-1}}{\Pochhammersym{-\gamma-n-k}{k-1}}		(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1))) = (pochhammer(- gamma - n - j, k - 1))/(pochhammer(- gamma - n - k, k - 1))	(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)]) == Divide[Pochhammer[- \[Gamma]- n - j, k - 1],Pochhammer[- \[Gamma]- n - k, k - 1]]	Failure	Failure	Failed [120 / 270] Result: .295470259e-1 Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 1} Result: .184734286e-1 Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 2} Result: .126342713e-1 Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 3} Result: .1117465202 Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 3, n = 1} ... skip entries to safe data	Failed [120 / 270] Result: Complex[0.004408174927732822, -0.03290306559789975] Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.003359414702213348, -0.020895843920590226] Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 2], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data