16.2: Difference between revisions

Latest revision as of 11:41, 28 June 2021

DLMF Formula Constraints Maple Mathematica Symbolic
Maple Symbolic
Mathematica Numeric
Maple Numeric
Mathematica

16.2.E3

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}

\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}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }

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))

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]]

Failure

Failed [258 / 300]

Result: .9712138727+.322304453e-1*I
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}

Result: -.6497511671-1.025183062*I
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}

... skip entries to safe data

Failed [258 / 300]

Result: Complex[0.9712138727144691, 0.032230445352325054]
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]]]}

Result: Complex[-0.6497511667213578, -1.0251830622105054]
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]]]}

... skip entries to safe data

16.2.E4

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}

\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}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle }

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))

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]]

Failure

Failed [258 / 300]

Result: .9712138726+.322304451e-1*I
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}

Result: 1.825190824+.5153748995*I
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}

... skip entries to safe data

Failed [258 / 300]

Result: Complex[0.9712138727144698, 0.03223044535232533]
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]]]}

Result: Complex[1.8251908240859445, 0.5153749002123968]
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]]]}

... skip entries to safe data

@@ Line 14: / Line 14: @@
 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
 |-
-| [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
+| [https://dlmf.nist.gov/16.2.E3 16.2.E3] || <math qid="Q5185">\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
 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
 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]
@@ Line 20: / Line 20: @@
 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>
 |-
-| [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
+| [https://dlmf.nist.gov/16.2.E4 16.2.E4] || <math qid="Q5186">\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
 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
 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]

16.2: Difference between revisions

Latest revision as of 11:41, 28 June 2021

Navigation menu

Search