Gamma Function - 5.18 -Gamma and -Beta Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.18.E1 ( a ; q ) n = ∏ k = 0 n - 1 ( 1 - a ⁒ q k ) q-Pochhammer-symbol π‘Ž π‘ž 𝑛 superscript subscript product π‘˜ 0 𝑛 1 1 π‘Ž superscript π‘ž π‘˜ {\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 ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁒ q k ) q-Pochhammer-symbol π‘Ž π‘ž superscript subscript product π‘˜ 0 1 π‘Ž superscript π‘ž π‘˜ {\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 Ξ“ q ⁑ ( z ) = ( q ; q ) ∞ ⁒ ( 1 - q ) 1 - z / ( q z ; q ) ∞ q-Gamma π‘ž 𝑧 q-Pochhammer-symbol π‘ž π‘ž superscript 1 π‘ž 1 𝑧 q-Pochhammer-symbol superscript π‘ž 𝑧 π‘ž {\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 Ξ“ q ⁑ ( 1 ) = Ξ“ q ⁑ ( 2 ) q-Gamma π‘ž 1 q-Gamma π‘ž 2 {\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 Ξ“ q ⁑ ( 2 ) = 1 q-Gamma π‘ž 2 1 {\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 n ⁒ \lxMathTweak ⁒ r ⁒ o ⁒ l ⁒ e = P ⁒ O ⁒ S ⁒ T ⁒ F ⁒ I ⁒ X ! q = Ξ“ q ⁑ ( n + 1 ) q-factorial 𝑛 π‘ž q-Gamma π‘ž 𝑛 1 {\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 Ξ“ q ⁑ ( z + 1 ) = 1 - q z 1 - q ⁒ Ξ“ q ⁑ ( z ) q-Gamma π‘ž 𝑧 1 1 superscript π‘ž 𝑧 1 π‘ž q-Gamma π‘ž 𝑧 {\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 Ξ“ q ⁑ ( x ) < Ξ“ r ⁑ ( x ) q-Gamma π‘ž π‘₯ q-Gamma π‘Ÿ π‘₯ {\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 Ξ“ q ⁑ ( x ) > Ξ“ r ⁑ ( x ) q-Gamma π‘ž π‘₯ q-Gamma π‘Ÿ π‘₯ {\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 lim q β†’ 1 - ⁑ Ξ“ q ⁑ ( z ) = Ξ“ ⁑ ( z ) subscript β†’ π‘ž limit-from 1 q-Gamma π‘ž 𝑧 Euler-Gamma 𝑧 {\displaystyle{\displaystyle\lim_{q\to 1-}\Gamma_{q}\left(z\right)=\Gamma\left% (z\right)}}
\lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z}		 β„œ ⁑ z > 0 𝑧 0 {\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
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