5.18

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.18.E1	${\displaystyle{\displaystyle\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})}}$ \qPochhammer{a}{q}{n} = \prod_{k=0}^{n-1}(1-aq^{k})		QPochhammer(a, q, n) = product(1 - a*(q)^(k), k = 0..n - 1)	QPochhammer[a, q, n] == Product[1 - a*(q)^(k), {k, 0, n - 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 60]
5.18.E3	${\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})}}$ \qPochhammer{a}{q}{\infty} = \prod_{k=0}^{\infty}(1-aq^{k})		QPochhammer(a, q, infinity) = product(1 - a*(q)^(k), k = 0..infinity)	QPochhammer[a, q, Infinity] == Product[1 - a*(q)^(k), {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [48 / 60] Result: Plus[Times[-1.0, QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]] Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
5.18.E4	${\displaystyle{\displaystyle\Gamma_{q}\left(z\right)=\left(q;q\right)_{\infty}(1-q)^{1-z}/\left(q^{z};q\right)_{\infty}}}$ \qGamma{q}@{z} = \qPochhammer{q}{q}{\infty}(1-q)^{1-z}/\qPochhammer{q^{z}}{q}{\infty}		QGAMMA(q, z) = QPochhammer(q, q, infinity)*(1 - q)^(1 - z)/QPochhammer((q)^(z), q, infinity)	QGamma[z,q] == QPochhammer[q, q, Infinity]*(1 - q)^(1 - z)/QPochhammer[(q)^(z), q, Infinity]	Error	Failure	-	Failed [56 / 70] Result: Plus[QGamma[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.4701974928403924, -0.07292434984262404], Power[QPochhammer[Complex[0.6918839380246471, 0.3371668184918191], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[QGamma[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.021172596861766507, 0.11798586945608598], Power[QPochhammer[Complex[0.6137803977754971, -0.16446196191399762], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
5.18.E5	${\displaystyle{\displaystyle\Gamma_{q}\left(1\right)=\Gamma_{q}\left(2\right)}}$ \qGamma{q}@{1} = \qGamma{q}@{2}		QGAMMA(q, 1) = QGAMMA(q, 2)	QGamma[1,q] == QGamma[2,q]	Error	Successful	-	Successful [Tested: 10]
5.18.E5	${\displaystyle{\displaystyle\Gamma_{q}\left(2\right)=1}}$ \qGamma{q}@{2} = 1		QGAMMA(q, 2) = 1	QGamma[2,q] == 1	Error	Successful	-	Successful [Tested: 10]
5.18.E6	${\displaystyle{\displaystyle n\lxMathTweak{role=POSTFIX}{\@nonfactorial_{q}}=\Gamma_{q}\left(n+1\right)}}$ \qfactorial{n}{q} = \qGamma{q}@{n+1}		QFactorial(n, q) = QGAMMA(q, n + 1)	QFactorial[n,q] == QGamma[n + 1,q]	Error	Successful	-	Successful [Tested: 30]
5.18.E7	${\displaystyle{\displaystyle\Gamma_{q}\left(z+1\right)=\frac{1-q^{z}}{1-q}\Gamma_{q}\left(z\right)}}$ \qGamma{q}@{z+1} = \frac{1-q^{z}}{1-q}\qGamma{q}@{z}		QGAMMA(q, z + 1) = (1 - (q)^(z))/(1 - q)*QGAMMA(q, z)	QGamma[z + 1,q] == Divide[1 - (q)^(z),1 - q]*QGamma[z,q]	Error	Failure	-	Failed [17 / 70] Result: Indeterminate Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
5.18.E8	${\displaystyle{\displaystyle\Gamma_{q}\left(x\right)<\Gamma_{r}\left(x\right)}}$ \qGamma{q}@{x} < \qGamma{r}@{x}		QGAMMA(q, x) < QGAMMA(r, x)	QGamma[x,q] < QGamma[x,r]	Failure	Failure	Error	Failed [174 / 180] Result: Less[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]} Result: Less[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]} ... skip entries to safe data
5.18.E9	${\displaystyle{\displaystyle\Gamma_{q}\left(x\right)>\Gamma_{r}\left(x\right)}}$ \qGamma{q}@{x} > \qGamma{r}@{x}		QGAMMA(q, x) > QGAMMA(r, x)	QGamma[x,q] > QGamma[x,r]	Failure	Failure	Error	Failed [174 / 180] Result: Greater[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]} Result: Greater[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]} ... skip entries to safe data
5.18.E10	${\displaystyle{\displaystyle\lim_{q\to 1-}\Gamma_{q}\left(z\right)=\Gamma\left(z\right)}}$ \lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z}	${\displaystyle{\displaystyle\Re z>0}}$	limit(QGAMMA(q, z), q = 1, left) = GAMMA(z)	Limit[QGamma[z,q], q -> 1, Direction -> "FromBelow", GenerateConditions->None] == Gamma[z]	Failure	Aborted	Error	Skipped - Because timed out