Functions of Number Theory - 27.14 Unrestricted Partitions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
27.14.E2	${\displaystyle{\displaystyle\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})}}$ \EulerPhi@{x} = \prod_{m=1}^{\infty}(1-x^{m})	${\displaystyle{\displaystyle|x|<1}}$	product(1-(x)^k, k = 1 .. infinity) = product(1 - (x)^(m), m = 1..infinity)	QPochhammer[x,x] == Product[1 - (x)^(m), {m, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
27.14.E3	${\displaystyle{\displaystyle\frac{1}{\mathit{f}\left(x\right)}=\sum_{n=0}^{\infty}p\left(n\right)x^{n}}}$ \frac{1}{\EulerPhi@{x}} = \sum_{n=0}^{\infty}\npartitions[]@{n}x^{n}	${\displaystyle{\displaystyle|x|<1}}$	(1)/(product(1-(x)^k, k = 1 .. infinity)) = sum(nops(partition(n))*(x)^(n), n = 0..infinity)	Error	Failure	Missing Macro Error	Error	-
27.14.E6	${\displaystyle{\displaystyle p\left(n\right)=\sum_{k=1}^{\infty}(-1)^{k+1}\left(p\left(n-\omega(k)\right)+p\left(n-\omega(-k)\right)\right)}}$ \npartitions[]@{n} = \sum_{k=1}^{\infty}(-1)^{k+1}\left(\npartitions[]@{n-\omega(k)}+\npartitions[]@{n-\omega(-k)}\right)		nops(partition(n)) = sum((- 1)^(k + 1)*(nops(partition(n - omega(k)))+ nops(partition(n - omega(- k)))), k = 1..infinity)	Error	Error	Missing Macro Error	-	-
27.14.E7	${\displaystyle{\displaystyle np\left(n\right)=\sum_{k=1}^{n}\sigma_{1}\left(k\right)p\left(n-k\right)}}$ n\npartitions[]@{n} = \sum_{k=1}^{n}\sumdivisors{1}@{k}\npartitions[]@{n-k}		n*nops(partition(n)) = sum(add(divisors(1))*nops(partition(n - k)), k = 1..n)	Error	Error	Missing Macro Error	-	-
27.14.E9	${\displaystyle{\displaystyle p\left(n\right)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_{k}(n)\*\left[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\sinh\left(\ifrac{K\sqrt{t}}{k}\right)}{\sqrt{t}}\right]_{t=n-(1/24)}}}$ \npartitions[]@{n} = \frac{1}{\cpi\sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_{k}(n)\*\left[\deriv{}{t}\frac{\sinh@{\ifrac{K\sqrt{t}}{k}}}{\sqrt{t}}\right]_{t=n-(1/24)}		nops(partition(n)) = (1)/(Pi*sqrt(2))*sum(sqrt(k)*A[k](n)*diff((sinh((K*sqrt(t))/(k)))/(sqrt(t)), t)[t = n -(1/24)], k = 1..infinity)	Error	Error	Missing Macro Error	-	-
27.14.E12	${\displaystyle{\displaystyle\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi\mathrm{i}n\tau})}}$ \Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\prod_{n=1}^{\infty}(1-e^{2\cpi\iunit n\tau})		Error	DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/12]*Product[1 - Exp[2*Pi*I*n*\[Tau]], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Successful [Tested: 1]
27.14.E13	${\displaystyle{\displaystyle\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\mathit{f}\left(e^{2\pi\mathrm{i}\tau}\right)}}$ \Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\EulerPhi@{e^{2\cpi\iunit\tau}}		Error	DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/12]*QPochhammer[Exp[2*Pi*I*\[Tau]],Exp[2*Pi*I*\[Tau]]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
27.14.E14	${\displaystyle{\displaystyle\eta\left(\frac{a\tau+b}{c\tau+d}\right)=\varepsilon(-\mathrm{i}(c\tau+d))^{\frac{1}{2}}\eta\left(\tau\right)}}$ \Dedekindeta@{\frac{a\tau+b}{c\tau+d}} = \varepsilon(-\iunit(c\tau+d))^{\frac{1}{2}}\Dedekindeta@{\tau}		Error	DedekindEta[Divide[a*\[Tau]+ b,c*\[Tau]+ d]] == \[CurlyEpsilon]*(- I*(c*\[Tau]+ d))^(Divide[1,2])* DedekindEta[\[Tau]]	Missing Macro Error	Failure	-	Failed [135 / 300] Result: Complex[0.13319594449577687, -0.32363546143707655] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, -2], Rule[ε, 1], Rule[τ, Complex[0, 1]]} Result: Complex[-0.41002146111087723, -1.4100702726503846] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, -2], Rule[ε, 2], Rule[τ, Complex[0, 1]]} ... skip entries to safe data
27.14.E15	${\displaystyle{\displaystyle 5\frac{(\mathit{f}\left(x^{5}\right))^{5}}{(\mathit{f}\left(x\right))^{6}}=\sum_{n=0}^{\infty}p\left(5n+4\right)x^{n}}}$ 5\frac{(\EulerPhi@{x^{5}})^{5}}{(\EulerPhi@{x})^{6}} = \sum_{n=0}^{\infty}\npartitions[]@{5n+4}x^{n}	${\displaystyle{\displaystyle|x|<1,|(x^{5})|<1}}$	5*((product(1-((x)^(5))^k, k = 1 .. infinity))^(5))/((product(1-(x)^k, k = 1 .. infinity))^(6)) = sum(nops(partition(5*n + 4))*(x)^(n), n = 0..infinity)	Error	Failure	Missing Macro Error	Error	-
27.14.E18	${\displaystyle{\displaystyle x\prod_{n=1}^{\infty}(1-x^{n})^{24}=\sum_{n=1}^{\infty}\tau\left(n\right)x^{n}}}$ x\prod_{n=1}^{\infty}(1-x^{n})^{24} = \sum_{n=1}^{\infty}\Ramanujantau@{n}x^{n}	${\displaystyle{\displaystyle|x|<1}}$	Error	x*Product[(1 - (x)^(n))^(24), {n, 1, Infinity}, GenerateConditions->None] == Sum[RamanujanTau[n]*(x)^(n), {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 1]