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| <div style="width: 100%; height: 75vh; overflow: auto;"> | | <div style="-moz-column-count:2; column-count:2;"> |
| {| class="wikitable sortable" style="margin: 0;"
| | ; Notation : [[16.1|16.1 Special Notation]]<br> |
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| | ; Generalized Hypergeometric Functions : [[16.2|16.2 Definition and Analytic Properties]]<br>[[16.3|16.3 Derivatives and Contiguous Functions]]<br>[[16.4|16.4 Argument Unity]]<br>[[16.5|16.5 Integral Representations and Integrals]]<br>[[16.6|16.6 Transformations of Variable]]<br>[[16.7|16.7 Relations to Other Functions]]<br>[[16.8|16.8 Differential Equations]]<br>[[16.9|16.9 Zeros]]<br>[[16.10|16.10 Expansions in Series of <math>\genhyperF{p}{q}</math> Functions]]<br>[[16.11|16.11 Asymptotic Expansions]]<br>[[16.12|16.12 Products]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | DLMF
| | ; Two-Variable Hypergeometric Functions : [[16.13|16.13 Appell Functions]]<br>[[16.14|16.14 Partial Differential Equations]]<br>[[16.15|16.15 Integral Representations and Integrals]]<br>[[16.16|16.16 Transformations of Variables]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Formula
| | ; Meijer <math>G</math> -Function : [[16.17|16.17 Definition]]<br>[[16.18|16.18 Special Cases]]<br>[[16.19|16.19 Identities]]<br>[[16.20|16.20 Integrals and Series]]<br>[[16.21|16.21 Differential Equation]]<br>[[16.22|16.22 Asymptotic Expansions]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Constraints
| | ; Applications : [[16.23|16.23 Mathematical Applications]]<br>[[16.24|16.24 Physical Applications]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Maple
| | ; Computation : [[16.25|16.25 Methods of Computation]]<br>[[16.26|16.26 Approximations]]<br>[[16.27|16.27 Software]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| | [https://dlmf.nist.gov/16.2.E3 16.2.E3] || [[Item:Q5185|<math>\genhyperF{p+1}{q}@@{-m,\mathbf{a}}{\mathbf{b}}{z} = \frac{\Pochhammersym{\mathbf{a}}{m}(-z)^{m}}{\Pochhammersym{\mathbf{b}}{m}}\genhyperF{q+1}{p}@@{-m,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q}}{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{p+1}{q}@@{-m,\mathbf{a}}{\mathbf{b}}{z} = \frac{\Pochhammersym{\mathbf{a}}{m}(-z)^{m}}{\Pochhammersym{\mathbf{b}}{m}}\genhyperF{q+1}{p}@@{-m,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q}}{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([- m , a], [b], z) = (pochhammer(a, m)*(- z)^(m))/(pochhammer(b, m))*hypergeom([- m , 1 - m - b], [1 - m - a], ((- 1)^(p + q))/(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{- m , a}, {b}, z] == Divide[Pochhammer[a, m]*(- z)^(m),Pochhammer[b, m]]*HypergeometricPFQ[{- m , 1 - m - b}, {1 - m - a}, Divide[(- 1)^(p + q),z]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [258 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9712138727+.322304453e-1*I
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| Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6497511671-1.025183062*I
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| Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [258 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9712138727144691, 0.032230445352325054]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.6497511667213578, -1.0251830622105054]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.2.E4 16.2.E4] || [[Item:Q5186|<math>\sum_{k=0}^{m}\frac{\Pochhammersym{\mathbf{a}}{k}}{\Pochhammersym{\mathbf{b}}{k}}\frac{z^{k}}{k!} = \frac{\Pochhammersym{\mathbf{a}}{m}z^{m}}{\Pochhammersym{\mathbf{b}}{m}m!}\genhyperF{q+2}{p}@@{-m,1,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q+1}}{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{m}\frac{\Pochhammersym{\mathbf{a}}{k}}{\Pochhammersym{\mathbf{b}}{k}}\frac{z^{k}}{k!} = \frac{\Pochhammersym{\mathbf{a}}{m}z^{m}}{\Pochhammersym{\mathbf{b}}{m}m!}\genhyperF{q+2}{p}@@{-m,1,1-m-\mathbf{b}}{1-m-\mathbf{a}}{\frac{(-1)^{p+q+1}}{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((pochhammer(a, k))/(pochhammer(b, k))*((z)^(k))/(factorial(k)), k = 0..m) = (pochhammer(a, m)*(z)^(m))/(pochhammer(b, m)*factorial(m))*hypergeom([- m , 1 , 1 - m - b], [1 - m - a], ((- 1)^(p + q + 1))/(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Divide[Pochhammer[a, k],Pochhammer[b, k]]*Divide[(z)^(k),(k)!], {k, 0, m}, GenerateConditions->None] == Divide[Pochhammer[a, m]*(z)^(m),Pochhammer[b, m]*(m)!]*HypergeometricPFQ[{- m , 1 , 1 - m - b}, {1 - m - a}, Divide[(- 1)^(p + q + 1),z]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [258 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9712138726+.322304451e-1*I
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| Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.825190824+.5153748995*I
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| Test Values: {a = -3/2, b = -3/2, p = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [258 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9712138727144698, 0.03223044535232533]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.8251908240859445, 0.5153749002123968]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.3.E5 16.3.E5] || [[Item:Q5192|<math>\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z*diff(z, z))^(n) = (z)^(n)* diff((z)^(n), [z$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z*D[z, z])^(n) == (z)^(n)* D[(z)^(n), {z, n}]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1616869430e-8-5.000000005*I
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| Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.000000005+.1616869430e-8*I
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| Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5000000000000001, -0.8660254037844386]
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| Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.0, -5.0]
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| Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.3.E6 16.3.E6] || [[Item:Q5193|<math>z\genhyperF{0}{1}@{-}{b+1}{z}+b(b-1)\genhyperF{0}{1}@{-}{b}{z}-b(b-1)\genhyperF{0}{1}@{-}{b-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\genhyperF{0}{1}@{-}{b+1}{z}+b(b-1)\genhyperF{0}{1}@{-}{b}{z}-b(b-1)\genhyperF{0}{1}@{-}{b-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*hypergeom([-], [b + 1], z)+ b*(b - 1)*hypergeom([-], [b], z)- b*(b - 1)*hypergeom([-], [b - 1], z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*HypergeometricPFQ[{-}, {b + 1}, z]+ b*(b - 1)*HypergeometricPFQ[{-}, {b}, z]- b*(b - 1)*HypergeometricPFQ[{-}, {b - 1}, z] == 0</syntaxhighlight> || Error || Failure || - || Error
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| | [https://dlmf.nist.gov/16.3.E7 16.3.E7] || [[Item:Q5194|<math>\genhyperF{3}{2}@@{a_{1}+2,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}(a_{1}+1)(1-z)+\genhyperF{3}{2}@@{a_{1}+1,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+\genhyperF{3}{2}@@{a_{1},a_{2},a_{3}}{b_{1},b_{2}}{z}\left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-\genhyperF{3}{2}@@{a_{1}-1,a_{2},a_{3}}{b_{1},b_{2}}{z}(a_{1}-b_{1})(a_{1}-b_{2}) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a_{1}+2,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}(a_{1}+1)(1-z)+\genhyperF{3}{2}@@{a_{1}+1,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+\genhyperF{3}{2}@@{a_{1},a_{2},a_{3}}{b_{1},b_{2}}{z}\left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-\genhyperF{3}{2}@@{a_{1}-1,a_{2},a_{3}}{b_{1},b_{2}}{z}(a_{1}-b_{1})(a_{1}-b_{2}) = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a[1]+ 2 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(a[1]+ 1)*(1 - z)+ hypergeom([a[1]+ 1 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(b[1]+ b[2]- 3*a[1]- 2 + z*(2*a[1]- a[2]- a[3]+ 1))+ hypergeom([a[1], a[2], a[3]], [b[1], b[2]], z)*((2*a[1]- b[1])*(2*a[1]- b[2])+ a[1]- (a[1])^(2)- z*(a[1]- a[2])*(a[1]- a[3]))- hypergeom([a[1]- 1 , a[2], a[3]], [b[1], b[2]], z)*(a[1]- b[1])*(a[1]- b[2]) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{Subscript[a, 1]+ 2 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[a, 1]+ 1)*(1 - z)+ HypergeometricPFQ[{Subscript[a, 1]+ 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[b, 1]+ Subscript[b, 2]- 3*Subscript[a, 1]- 2 + z*(2*Subscript[a, 1]- Subscript[a, 2]- Subscript[a, 3]+ 1))+ HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*((2*Subscript[a, 1]- Subscript[b, 1])*(2*Subscript[a, 1]- Subscript[b, 2])+ Subscript[a, 1]- (Subscript[a, 1])^(2)- z*(Subscript[a, 1]- Subscript[a, 2])*(Subscript[a, 1]- Subscript[a, 3]))- HypergeometricPFQ[{Subscript[a, 1]- 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*(Subscript[a, 1]- Subscript[b, 1])*(Subscript[a, 1]- Subscript[b, 2]) == 0</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.7372028395654344, 0.5250871122698257], Times[Complex[-0.5000000000000001, -0.8660254037844386], a]]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[4.427028162877593, -11.419461015230842], Times[Complex[-0.5000000000000001, -0.8660254037844386], a]]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.4.E1 16.4.E1] || [[Item:Q5195|<math>a_{q}+b_{q} = a_{q+1}+1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{q}+b_{q} = a_{q+1}+1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[q]+ b[q] = a[q + 1]+ 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, q]+ Subscript[b, q] == Subscript[a, q + 1]+ 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/16.4.E3 16.4.E3] || [[Item:Q5197|<math>\genhyperF{3}{2}@@{-n,a,b}{c,d}{1} = \frac{\Pochhammersym{c-a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}\Pochhammersym{c-a-b}{n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{-n,a,b}{c,d}{1} = \frac{\Pochhammersym{c-a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}\Pochhammersym{c-a-b}{n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([- n , a , b], [c , d], 1) = (pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n)*pochhammer(c - a - b, n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{- n , a , b}, {c , d}, 1] == Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]*Pochhammer[c - a - b, n]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [281 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.299038106-.7499999997*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.872595264-1.774519052*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [281 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.299038105676658, -0.7499999999999998]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.872595264191645, -1.7745190528383286]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.4.E4 16.4.E4] || [[Item:Q5198|<math>\genhyperF{3}{2}@@{a,b,c}{a-b+1,a-c+1}{1} = \frac{\EulerGamma@{\frac{1}{2}a+1}\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{\frac{1}{2}a-b-c+1}}{\EulerGamma@{a+1}\EulerGamma@{\frac{1}{2}a-b+1}\EulerGamma@{\frac{1}{2}a-c+1}\EulerGamma@{a-b-c+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,b,c}{a-b+1,a-c+1}{1} = \frac{\EulerGamma@{\frac{1}{2}a+1}\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{\frac{1}{2}a-b-c+1}}{\EulerGamma@{a+1}\EulerGamma@{\frac{1}{2}a-b+1}\EulerGamma@{\frac{1}{2}a-c+1}\EulerGamma@{a-b-c+1}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}a+1)} > 0, \realpart@@{(a-b+1)} > 0, \realpart@@{(a-c+1)} > 0, \realpart@@{(\frac{1}{2}a-b-c+1)} > 0, \realpart@@{(a+1)} > 0, \realpart@@{(\frac{1}{2}a-b+1)} > 0, \realpart@@{(\frac{1}{2}a-c+1)} > 0, \realpart@@{(a-b-c+1)} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a , b , c], [a - b + 1 , a - c + 1], 1) = (GAMMA((1)/(2)*a + 1)*GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA((1)/(2)*a - b - c + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1)*GAMMA((1)/(2)*a - c + 1)*GAMMA(a - b - c + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b , c}, {a - b + 1 , a - c + 1}, 1] == Divide[Gamma[Divide[1,2]*a + 1]*Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[Divide[1,2]*a - b - c + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]*Gamma[Divide[1,2]*a - c + 1]*Gamma[a - b - c + 1]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 69]
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| | [https://dlmf.nist.gov/16.4.E6 16.4.E6] || [[Item:Q5200|<math>\genhyperF{3}{2}@@{a,b,c}{\frac{1}{2}(a+b+1),2c}{1} = \frac{\EulerGamma@{\frac{1}{2}}\EulerGamma@{c+\frac{1}{2}}\EulerGamma@{\frac{1}{2}(a+b+1)}\EulerGamma@{c+\frac{1}{2}(1-a-b)}}{\EulerGamma@{\frac{1}{2}(a+1)}\EulerGamma@{\frac{1}{2}(b+1)}\EulerGamma@{c+\frac{1}{2}(1-a)}\EulerGamma@{c+\frac{1}{2}(1-b)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,b,c}{\frac{1}{2}(a+b+1),2c}{1} = \frac{\EulerGamma@{\frac{1}{2}}\EulerGamma@{c+\frac{1}{2}}\EulerGamma@{\frac{1}{2}(a+b+1)}\EulerGamma@{c+\frac{1}{2}(1-a-b)}}{\EulerGamma@{\frac{1}{2}(a+1)}\EulerGamma@{\frac{1}{2}(b+1)}\EulerGamma@{c+\frac{1}{2}(1-a)}\EulerGamma@{c+\frac{1}{2}(1-b)}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2})} > 0, \realpart@@{(c+\frac{1}{2})} > 0, \realpart@@{(\frac{1}{2}(a+b+1))} > 0, \realpart@@{(c+\frac{1}{2}(1-a-b))} > 0, \realpart@@{(\frac{1}{2}(a+1))} > 0, \realpart@@{(\frac{1}{2}(b+1))} > 0, \realpart@@{(c+\frac{1}{2}(1-a))} > 0, \realpart@@{(c+\frac{1}{2}(1-b))} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a , b , c], [(1)/(2)*(a + b + 1), 2*c], 1) = (GAMMA((1)/(2))*GAMMA(c +(1)/(2))*GAMMA((1)/(2)*(a + b + 1))*GAMMA(c +(1)/(2)*(1 - a - b)))/(GAMMA((1)/(2)*(a + 1))*GAMMA((1)/(2)*(b + 1))*GAMMA(c +(1)/(2)*(1 - a))*GAMMA(c +(1)/(2)*(1 - b)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b , c}, {Divide[1,2]*(a + b + 1), 2*c}, 1] == Divide[Gamma[Divide[1,2]]*Gamma[c +Divide[1,2]]*Gamma[Divide[1,2]*(a + b + 1)]*Gamma[c +Divide[1,2]*(1 - a - b)],Gamma[Divide[1,2]*(a + 1)]*Gamma[Divide[1,2]*(b + 1)]*Gamma[c +Divide[1,2]*(1 - a)]*Gamma[c +Divide[1,2]*(1 - b)]]</syntaxhighlight> || Successful || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.4.E7 16.4.E7] || [[Item:Q5201|<math>\genhyperF{3}{2}@@{a,1-a,c}{d,2c-d+1}{1} = \frac{\pi\EulerGamma@{d}\EulerGamma@{2c-d+1}2^{1-2c}}{\EulerGamma@{c+\frac{1}{2}(a-d+1)}\EulerGamma@{c+1-\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(d-a+1)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,1-a,c}{d,2c-d+1}{1} = \frac{\pi\EulerGamma@{d}\EulerGamma@{2c-d+1}2^{1-2c}}{\EulerGamma@{c+\frac{1}{2}(a-d+1)}\EulerGamma@{c+1-\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(d-a+1)}}</syntaxhighlight> || <math>\realpart@@{d} > 0, \realpart@@{(2c-d+1)} > 0, \realpart@@{(c+\frac{1}{2}(a-d+1))} > 0, \realpart@@{(c+1-\frac{1}{2}(a+d))} > 0, \realpart@@{(\frac{1}{2}(a+d))} > 0, \realpart@@{(\frac{1}{2}(d-a+1))} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a , 1 - a , c], [d , 2*c - d + 1], 1) = (Pi*GAMMA(d)*GAMMA(2*c - d + 1)*(2)^(1 - 2*c))/(GAMMA(c +(1)/(2)*(a - d + 1))*GAMMA(c + 1 -(1)/(2)*(a + d))*GAMMA((1)/(2)*(a + d))*GAMMA((1)/(2)*(d - a + 1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , 1 - a , c}, {d , 2*c - d + 1}, 1] == Divide[Pi*Gamma[d]*Gamma[2*c - d + 1]*(2)^(1 - 2*c),Gamma[c +Divide[1,2]*(a - d + 1)]*Gamma[c + 1 -Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(d - a + 1)]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 40]
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| | [https://dlmf.nist.gov/16.4.E8 16.4.E8] || [[Item:Q5202|<math>\genhyperF{3}{2}@@{-n,a,1-a}{d,1-d-2n}{1} = \frac{\Pochhammersym{\frac{1}{2}(a+d)}{n}\Pochhammersym{\frac{1}{2}(d-a+1)}{n}}{\Pochhammersym{\frac{1}{2}d}{n}\Pochhammersym{\frac{1}{2}(d+1)}{n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{-n,a,1-a}{d,1-d-2n}{1} = \frac{\Pochhammersym{\frac{1}{2}(a+d)}{n}\Pochhammersym{\frac{1}{2}(d-a+1)}{n}}{\Pochhammersym{\frac{1}{2}d}{n}\Pochhammersym{\frac{1}{2}(d+1)}{n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([- n , a , 1 - a], [d , 1 - d - 2*n], 1) = (pochhammer((1)/(2)*(a + d), n)*pochhammer((1)/(2)*(d - a + 1), n))/(pochhammer((1)/(2)*d, n)*pochhammer((1)/(2)*(d + 1), n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{- n , a , 1 - a}, {d , 1 - d - 2*n}, 1] == Divide[Pochhammer[Divide[1,2]*(a + d), n]*Pochhammer[Divide[1,2]*(d - a + 1), n],Pochhammer[Divide[1,2]*d, n]*Pochhammer[Divide[1,2]*(d + 1), n]]</syntaxhighlight> || Failure || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [112 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5976759376684342, 0.11432617133831768]
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| Test Values: {Rule[a, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.4201764035832656, 0.019572796644155455]
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| Test Values: {Rule[a, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.4.E9 16.4.E9] || [[Item:Q5203|<math>\genhyperF{5}{4}@@{a,\frac{1}{2}a+1,b,c,d}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1}{1} = \frac{\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{a-d+1}\EulerGamma@{a-b-c-d+1}}{\EulerGamma@{a+1}\EulerGamma@{a-b-c+1}\EulerGamma@{a-b-d+1}\EulerGamma@{a-c-d+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{5}{4}@@{a,\frac{1}{2}a+1,b,c,d}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1}{1} = \frac{\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{a-d+1}\EulerGamma@{a-b-c-d+1}}{\EulerGamma@{a+1}\EulerGamma@{a-b-c+1}\EulerGamma@{a-b-d+1}\EulerGamma@{a-c-d+1}}</syntaxhighlight> || <math>\realpart@@{(a-b+1)} > 0, \realpart@@{(a-c+1)} > 0, \realpart@@{(a-d+1)} > 0, \realpart@@{(a-b-c-d+1)} > 0, \realpart@@{(a+1)} > 0, \realpart@@{(a-b-c+1)} > 0, \realpart@@{(a-b-d+1)} > 0, \realpart@@{(a-c-d+1)} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a ,(1)/(2)*a + 1 , b , c , d], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1], 1) = (GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA(a - d + 1)*GAMMA(a - b - c - d + 1))/(GAMMA(a + 1)*GAMMA(a - b - c + 1)*GAMMA(a - b - d + 1)*GAMMA(a - c - d + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1}, 1] == Divide[Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[a - d + 1]*Gamma[a - b - c - d + 1],Gamma[a + 1]*Gamma[a - b - c + 1]*Gamma[a - b - d + 1]*Gamma[a - c - d + 1]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/16.4.E10 16.4.E10] || [[Item:Q5204|<math>\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,f,-n}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1}{1} = \frac{\Pochhammersym{a+1}{n}\Pochhammersym{a-b-c+1}{n}\Pochhammersym{a-b-d+1}{n}\Pochhammersym{a-c-d+1}{n}}{\Pochhammersym{a-b+1}{n}\Pochhammersym{a-c+1}{n}\Pochhammersym{a-d+1}{n}\Pochhammersym{a-b-c-d+1}{n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,f,-n}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1}{1} = \frac{\Pochhammersym{a+1}{n}\Pochhammersym{a-b-c+1}{n}\Pochhammersym{a-b-d+1}{n}\Pochhammersym{a-c-d+1}{n}}{\Pochhammersym{a-b+1}{n}\Pochhammersym{a-c+1}{n}\Pochhammersym{a-d+1}{n}\Pochhammersym{a-b-c-d+1}{n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a ,(1)/(2)*a + 1 , b , c , d , f , - n], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1], 1) = (pochhammer(a + 1, n)*pochhammer(a - b - c + 1, n)*pochhammer(a - b - d + 1, n)*pochhammer(a - c - d + 1, n))/(pochhammer(a - b + 1, n)*pochhammer(a - c + 1, n)*pochhammer(a - d + 1, n)*pochhammer(a - b - c - d + 1, n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , f , - n}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1}, 1] == Divide[Pochhammer[a + 1, n]*Pochhammer[a - b - c + 1, n]*Pochhammer[a - b - d + 1, n]*Pochhammer[a - c - d + 1, n],Pochhammer[a - b + 1, n]*Pochhammer[a - c + 1, n]*Pochhammer[a - d + 1, n]*Pochhammer[a - b - c - d + 1, n]]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2096832772+.6841105627e-1*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, f = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1072644549-.5307589441*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, f = -1/2+1/2*I*3^(1/2), n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.4.E11 16.4.E11] || [[Item:Q5205|<math>\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \frac{\EulerGamma@{e}\EulerGamma@{d+e-a-b-c}}{\EulerGamma@{e-a}\EulerGamma@{d+e-b-c}}\genhyperF{3}{2}@@{a,d-b,d-c}{d,d+e-b-c}{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \frac{\EulerGamma@{e}\EulerGamma@{d+e-a-b-c}}{\EulerGamma@{e-a}\EulerGamma@{d+e-b-c}}\genhyperF{3}{2}@@{a,d-b,d-c}{d,d+e-b-c}{1}</syntaxhighlight> || <math>\realpart@@{e} > 0, \realpart@@{(d+e-a-b-c)} > 0, \realpart@@{(e-a)} > 0, \realpart@@{(d+e-b-c)} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a , b , c], [d , e], 1) = (GAMMA(e)*GAMMA(d + e - a - b - c))/(GAMMA(e - a)*GAMMA(d + e - b - c))*hypergeom([a , d - b , d - c], [d , d + e - b - c], 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b , c}, {d , e}, 1] == Divide[Gamma[e]*Gamma[d + e - a - b - c],Gamma[e - a]*Gamma[d + e - b - c]]*HypergeometricPFQ[{a , d - b , d - c}, {d , d + e - b - c}, 1]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.4.E12 16.4.E12] || [[Item:Q5206|<math>(a-d)(b-d)(c-d)\left(\genhyperF{3}{2}@@{a,b,c}{d+1,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d,e}{1}\right)+abc\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = d(d-1)(a+b+c-d-e+1)\left(\genhyperF{3}{2}@@{a,b,c}{d,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d-1,e}{1}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a-d)(b-d)(c-d)\left(\genhyperF{3}{2}@@{a,b,c}{d+1,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d,e}{1}\right)+abc\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = d(d-1)(a+b+c-d-e+1)\left(\genhyperF{3}{2}@@{a,b,c}{d,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d-1,e}{1}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(a - d)*(b - d)*(c - d)*(hypergeom([a , b , c], [d + 1 , e], 1)- hypergeom([a , b , c], [d , e], 1))+ a*b*c*hypergeom([a , b , c], [d , e], 1) = d*(d - 1)*(a + b + c - d - e + 1)*(hypergeom([a , b , c], [d , e], 1)- hypergeom([a , b , c], [d - 1 , e], 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a - d)*(b - d)*(c - d)*(HypergeometricPFQ[{a , b , c}, {d + 1 , e}, 1]- HypergeometricPFQ[{a , b , c}, {d , e}, 1])+ a*b*c*HypergeometricPFQ[{a , b , c}, {d , e}, 1] == d*(d - 1)*(a + b + c - d - e + 1)*(HypergeometricPFQ[{a , b , c}, {d , e}, 1]- HypergeometricPFQ[{a , b , c}, {d - 1 , e}, 1])</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.4.E13 16.4.E13] || [[Item:Q5207|<math>\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \dfrac{c(e-a)}{de}\genhyperF{3}{2}@@{a,b+1,c+1}{d+1,e+1}{1}+\dfrac{d-c}{d}\genhyperF{3}{2}@@{a,b+1,c}{d+1,e}{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \dfrac{c(e-a)}{de}\genhyperF{3}{2}@@{a,b+1,c+1}{d+1,e+1}{1}+\dfrac{d-c}{d}\genhyperF{3}{2}@@{a,b+1,c}{d+1,e}{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a , b , c], [d , e], 1) = (c*(e - a))/(d*e)*hypergeom([a , b + 1 , c + 1], [d + 1 , e + 1], 1)+(d - c)/(d)*hypergeom([a , b + 1 , c], [d + 1 , e], 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b , c}, {d , e}, 1] == Divide[c*(e - a),d*e]*HypergeometricPFQ[{a , b + 1 , c + 1}, {d + 1 , e + 1}, 1]+Divide[d - c,d]*HypergeometricPFQ[{a , b + 1 , c}, {d + 1 , e}, 1]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.4.E14 16.4.E14] || [[Item:Q5208|<math>\genhyperF{4}{3}@@{-n,a,b,c}{d,e,f}{1} = \frac{\Pochhammersym{e-a}{n}\Pochhammersym{f-a}{n}}{\Pochhammersym{e}{n}\Pochhammersym{f}{n}}\genhyperF{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{4}{3}@@{-n,a,b,c}{d,e,f}{1} = \frac{\Pochhammersym{e-a}{n}\Pochhammersym{f-a}{n}}{\Pochhammersym{e}{n}\Pochhammersym{f}{n}}\genhyperF{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([- n , a , b , c], [d , e , f], 1) = (pochhammer(e - a, n)*pochhammer(f - a, n))/(pochhammer(e, n)*pochhammer(f, n))*hypergeom([- n , a , d - b , d - c], [d , a - e - n + 1 , a - f - n + 1], 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{- n , a , b , c}, {d , e , f}, 1] == Divide[Pochhammer[e - a, n]*Pochhammer[f - a, n],Pochhammer[e, n]*Pochhammer[f, n]]*HypergeometricPFQ[{- n , a , d - b , d - c}, {d , a - e - n + 1 , a - f - n + 1}, 1]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-7.272114317029979, 8.095671475544961]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-18.740982240718687, 40.16393590217987]
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| Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.4.E15 16.4.E15] || [[Item:Q5209|<math>\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,e,f}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1}{1} = \frac{\EulerGamma@{a-d+1}\EulerGamma@{a-e+1}\EulerGamma@{a-f+1}\EulerGamma@{a-d-e-f+1}}{\EulerGamma@{a+1}\EulerGamma@{a-d-e+1}\EulerGamma@{a-d-f+1}\EulerGamma@{a-e-f+1}}\genhyperF{4}{3}@@{a-b-c+1,d,e,f}{a-b+1,a-c+1,d+e+f-a}{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,e,f}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1}{1} = \frac{\EulerGamma@{a-d+1}\EulerGamma@{a-e+1}\EulerGamma@{a-f+1}\EulerGamma@{a-d-e-f+1}}{\EulerGamma@{a+1}\EulerGamma@{a-d-e+1}\EulerGamma@{a-d-f+1}\EulerGamma@{a-e-f+1}}\genhyperF{4}{3}@@{a-b-c+1,d,e,f}{a-b+1,a-c+1,d+e+f-a}{1}</syntaxhighlight> || <math>\realpart@@{(a-d+1)} > 0, \realpart@@{(a-e+1)} > 0, \realpart@@{(a-f+1)} > 0, \realpart@@{(a-d-e-f+1)} > 0, \realpart@@{(a+1)} > 0, \realpart@@{(a-d-e+1)} > 0, \realpart@@{(a-d-f+1)} > 0, \realpart@@{(a-e-f+1)} > 0</math> || <syntaxhighlight lang=mathematica>hypergeom([a ,(1)/(2)*a + 1 , b , c , d , e , f], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1], 1) = (GAMMA(a - d + 1)*GAMMA(a - e + 1)*GAMMA(a - f + 1)*GAMMA(a - d - e - f + 1))/(GAMMA(a + 1)*GAMMA(a - d - e + 1)*GAMMA(a - d - f + 1)*GAMMA(a - e - f + 1))*hypergeom([a - b - c + 1 , d , e , f], [a - b + 1 , a - c + 1 , d + e + f - a], 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , e , f}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1}, 1] == Divide[Gamma[a - d + 1]*Gamma[a - e + 1]*Gamma[a - f + 1]*Gamma[a - d - e - f + 1],Gamma[a + 1]*Gamma[a - d - e + 1]*Gamma[a - d - f + 1]*Gamma[a - e - f + 1]]*HypergeometricPFQ[{a - b - c + 1 , d , e , f}, {a - b + 1 , a - c + 1 , d + e + f - a}, 1]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.6.E1 16.6.E1] || [[Item:Q5216|<math>\genhyperF{3}{2}@@{a,b,c}{a-b+1,a-c+1}{z} = (1-z)^{-a}\genhyperF{3}{2}@@{a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)}{a-b+1,a-c+1}{\frac{-4z}{(1-z)^{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,b,c}{a-b+1,a-c+1}{z} = (1-z)^{-a}\genhyperF{3}{2}@@{a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)}{a-b+1,a-c+1}{\frac{-4z}{(1-z)^{2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a , b , c], [a - b + 1 , a - c + 1], z) = (1 - z)^(- a)* hypergeom([a - b - c + 1 ,(1)/(2)*a ,(1)/(2)*(a + 1)], [a - b + 1 , a - c + 1], (- 4*z)/((1 - z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , b , c}, {a - b + 1 , a - c + 1}, z] == (1 - z)^(- a)* HypergeometricPFQ[{a - b - c + 1 ,Divide[1,2]*a ,Divide[1,2]*(a + 1)}, {a - b + 1 , a - c + 1}, Divide[- 4*z,(1 - z)^(2)]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [258 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.076719790+.860205503*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.428233246+.1e-8*I
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/16.6.E2 16.6.E2] || [[Item:Q5217|<math>\genhyperF{3}{2}@@{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}}{\frac{z}{4}} = (1-z)^{-a}\genhyperF{3}{2}@@{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}}{\frac{-27z}{4(1-z)^{3}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}}{\frac{z}{4}} = (1-z)^{-a}\genhyperF{3}{2}@@{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}}{\frac{-27z}{4(1-z)^{3}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([a , 2*b - a - 1 , 2 - 2*b + a], [b , a - b +(3)/(2)], (z)/(4)) = (1 - z)^(- a)* hypergeom([(1)/(3)*a ,(1)/(3)*a +(1)/(3),(1)/(3)*a +(2)/(3)], [b , a - b +(3)/(2)], (- 27*z)/(4*(1 - z)^(3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{a , 2*b - a - 1 , 2 - 2*b + a}, {b , a - b +Divide[3,2]}, Divide[z,4]] == (1 - z)^(- a)* HypergeometricPFQ[{Divide[1,3]*a ,Divide[1,3]*a +Divide[1,3],Divide[1,3]*a +Divide[2,3]}, {b , a - b +Divide[3,2]}, Divide[- 27*z,4*(1 - z)^(3)]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [216 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1888061791+.200959324e-1*I
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| Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.140210603-.95166922e-1*I
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| Test Values: {a = -3/2, b = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/16.8.E4 16.8.E4] || [[Item:Q5222|<math>z^{q}D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z^{q}D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w = 0</syntaxhighlight> || <math>p \leq q</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(z)^(q)* (D)^(q + 1)* w + sum((z)^(j - 1)*(alpha[j]*z + beta[j])*(D)^(j)* w , j = 1..q)+ alpha[0]*w = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(z)^(q)* (D)^(q + 1)* w + Sum[(z)^(j - 1)*(Subscript[\[Alpha], j]*z + Subscript[\[Beta], j])*(D)^(j)* w , {j, 1, q}, GenerateConditions->None]+ Subscript[\[Alpha], 0]*w == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/16.8.E5 16.8.E5] || [[Item:Q5223|<math>z^{q}(1-z)D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z^{q}(1-z)D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w = 0</syntaxhighlight> || <math>p = q+1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(z)^(q)*(1 - z)*(D)^(q + 1)* w + sum((z)^(j - 1)*(alpha[j]*z + beta[j])*(D)^(j)* w , j = 1..q)+ alpha[0]*w = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(z)^(q)*(1 - z)*(D)^(q + 1)* w + Sum[(z)^(j - 1)*(Subscript[\[Alpha], j]*z + Subscript[\[Beta], j])*(D)^(j)* w , {j, 1, q}, GenerateConditions->None]+ Subscript[\[Alpha], 0]*w == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/16.11#Ex3 16.11#Ex3] || [[Item:Q5237|<math>c_{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>c_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">c[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[c, 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/16.12.E1 16.12.E1] || [[Item:Q5246|<math>\genhyperF{0}{1}@{-}{a}{z}\genhyperF{0}{1}@{-}{b}{z} = \genhyperF{2}{3}@@{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1}{4z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{0}{1}@{-}{a}{z}\genhyperF{0}{1}@{-}{b}{z} = \genhyperF{2}{3}@@{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1}{4z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([-], [a], z)*hypergeom([-], [b], z) = hypergeom([(1)/(2)*(a + b),(1)/(2)*(a + b - 1)], [a , b , a + b - 1], 4*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{-}, {a}, z]*HypergeometricPFQ[{-}, {b}, z] == HypergeometricPFQ[{Divide[1,2]*(a + b),Divide[1,2]*(a + b - 1)}, {a , b , a + b - 1}, 4*z]</syntaxhighlight> || Error || Failure || - || Error
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| |-
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| | [https://dlmf.nist.gov/16.12.E2 16.12.E2] || [[Item:Q5247|<math>\left(\genhyperF{2}{1}@@{a,b}{a+b+\frac{1}{2}}{z}\right)^{2} = \genhyperF{3}{2}@@{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\genhyperF{2}{1}@@{a,b}{a+b+\frac{1}{2}}{z}\right)^{2} = \genhyperF{3}{2}@@{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(hypergeom([a , b], [a + b +(1)/(2)], z))^(2) = hypergeom([2*a , 2*b , a + b], [a + b +(1)/(2), 2*a + 2*b], z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(HypergeometricPFQ[{a , b}, {a + b +Divide[1,2]}, z])^(2) == HypergeometricPFQ[{2*a , 2*b , a + b}, {a + b +Divide[1,2], 2*a + 2*b}, z]</syntaxhighlight> || Failure || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [108 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.205771365940054, 0.2846096908265261]
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| Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-23.50000000000001, -28.578838324886455]
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| Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/16.12.E3 16.12.E3] || [[Item:Q5248|<math>\left(\genhyperF{2}{1}@@{a,b}{c}{z}\right)^{2} = \sum_{k=0}^{\infty}\frac{\Pochhammersym{2a}{k}\Pochhammersym{2b}{k}\Pochhammersym{c-\frac{1}{2}}{k}}{\Pochhammersym{c}{k}\Pochhammersym{2c-1}{k}k!}\genhyperF{4}{3}@@{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c}{1}z^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\genhyperF{2}{1}@@{a,b}{c}{z}\right)^{2} = \sum_{k=0}^{\infty}\frac{\Pochhammersym{2a}{k}\Pochhammersym{2b}{k}\Pochhammersym{c-\frac{1}{2}}{k}}{\Pochhammersym{c}{k}\Pochhammersym{2c-1}{k}k!}\genhyperF{4}{3}@@{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c}{1}z^{k}</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>(hypergeom([a , b], [c], z))^(2) = sum((pochhammer(2*a, k)*pochhammer(2*b, k)*pochhammer(c -(1)/(2), k))/(pochhammer(c, k)*pochhammer(2*c - 1, k)*factorial(k))*hypergeom([-(1)/(2)*k ,(1)/(2)*(1 - k), a + b - c +(1)/(2),(1)/(2)], [a +(1)/(2), b +(1)/(2),(3)/(2)- k - c], 1)*(z)^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(HypergeometricPFQ[{a , b}, {c}, z])^(2) == Sum[Divide[Pochhammer[2*a, k]*Pochhammer[2*b, k]*Pochhammer[c -Divide[1,2], k],Pochhammer[c, k]*Pochhammer[2*c - 1, k]*(k)!]*HypergeometricPFQ[{-Divide[1,2]*k ,Divide[1,2]*(1 - k), a + b - c +Divide[1,2],Divide[1,2]}, {a +Divide[1,2], b +Divide[1,2],Divide[3,2]- k - c}, 1]*(z)^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [159 / 216]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1250000000
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| Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 5.872053804
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| Test Values: {a = -3/2, b = -3/2, c = -1/2, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| |-
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| | [https://dlmf.nist.gov/16.16.E5 16.16.E5] || [[Item:Q5271|<math>\AppellF{3}@{\alpha}{\gamma-\alpha}{\beta}{\gamma-\beta}{\gamma}{x}{y} = (1-y)^{\alpha+\beta-\gamma}\genhyperF{2}{1}@@{\alpha,\beta}{\gamma}{x+y-xy}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\AppellF{3}@{\alpha}{\gamma-\alpha}{\beta}{\gamma-\beta}{\gamma}{x}{y} = (1-y)^{\alpha+\beta-\gamma}\genhyperF{2}{1}@@{\alpha,\beta}{\gamma}{x+y-xy}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>AppellF[3, , \[Alpha], \[Gamma]- \[Alpha], \[Beta], \[Gamma]- \[Beta]]*\[Gamma]*x*y == (1 - y)^(\[Alpha]+ \[Beta]- \[Gamma])* HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Gamma]}, x + y - x*y]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.33907796278424684, 2.1694931088262193], Times[Complex[-1.948557158514987, -1.1249999999999998], AppellF[3.0, Null, 1.5, Complex[-0.6339745962155613, 0.49999999999999994], 1.5, Complex[-0.6339745962155613, 0.49999999999999994]]]]
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| Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[3.1592684418872854, 2.774129956365469], Times[Complex[1.1249999999999996, -1.948557158514987], AppellF[3.0, Null, 1.5, Complex[-1.9999999999999998, 0.8660254037844387], 1.5, Complex[-1.9999999999999998, 0.8660254037844387]]]]
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| Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
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| | [https://dlmf.nist.gov/16.16.E6 16.16.E6] || [[Item:Q5272|<math>\AppellF{4}@{\alpha}{\beta}{\gamma}{\alpha+\beta-\gamma+1}{x(1-y)}{y(1-x)} = \genhyperF{2}{1}@@{\alpha,\beta}{\gamma}{x}\genhyperF{2}{1}@@{\alpha,\beta}{\alpha+\beta-\gamma+1}{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\AppellF{4}@{\alpha}{\beta}{\gamma}{\alpha+\beta-\gamma+1}{x(1-y)}{y(1-x)} = \genhyperF{2}{1}@@{\alpha,\beta}{\gamma}{x}\genhyperF{2}{1}@@{\alpha,\beta}{\alpha+\beta-\gamma+1}{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>AppellF[4, , \[Alpha], \[Beta], \[Gamma], \[Alpha]+ \[Beta]- \[Gamma]+ 1]*x*(1 - y)*y*(1 - x) == HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Gamma]}, x]*HypergeometricPFQ[{\[Alpha], \[Beta]}, {\[Alpha]+ \[Beta]- \[Gamma]+ 1}, y]</syntaxhighlight> || Missing Macro Error || Failure || - || Skip - No test values generated
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| |-
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| | [https://dlmf.nist.gov/16.23.E1 16.23.E1] || [[Item:Q5289|<math>\genhyperF{3}{2}@@{-n,n+\alpha+2,\frac{1}{2}(\alpha+1)}{\alpha+1,\frac{1}{2}(\alpha+3)}{x} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genhyperF{3}{2}@@{-n,n+\alpha+2,\frac{1}{2}(\alpha+1)}{\alpha+1,\frac{1}{2}(\alpha+3)}{x} > 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([- n , n + alpha + 2 ,(1)/(2)*(alpha + 1)], [alpha + 1 ,(1)/(2)*(alpha + 3)], x) > 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{- n , n + \[Alpha]+ 2 ,Divide[1,2]*(\[Alpha]+ 1)}, {\[Alpha]+ 1 ,Divide[1,2]*(\[Alpha]+ 3)}, x] > 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0. < -.5000000000
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| Test Values: {alpha = 3/2, x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0. < -1.482142857
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| Test Values: {alpha = 3/2, x = 3/2, n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
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| Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
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| Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[α, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |}
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| </div> | | </div> |