Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Hypergeometric2F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Specific values > Specialized values > For fixed z and with symbolical integers in parameters > For fixed z and a=1, b=n, c=p





http://functions.wolfram.com/07.23.03.0175.01









  


  








Input Form





Hypergeometric2F1[1, n, m, z] == ((m - 1)!/((m - n - 1)! z)) (Sum[((m - n - k - 1)!/(m - k - 1)!) ((z - 1)/z)^(k - 1), {k, 1, m - n - 1}] - (z/(n - 1)!) ((z - 1)/z)^(m - n - 1) (Sum[1/(z^k (n - k)), {k, 1, n - 1}] + Log[1 - z]/z^n)) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0 && n < m








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "n", ",", "m", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "1"]], ")"]], "!"]], "z"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k", "-", "1"]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "z"], ")"]], RowBox[List["k", "-", "1"]]]]]]], "-", RowBox[List[FractionBox["z", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "z"], ")"]], RowBox[List["m", "-", "n", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[SuperscriptBox["z", RowBox[List["-", "k"]]], RowBox[List["n", "-", "k"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["n", "<", "m"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> n </mi> </mrow> <mo> ; </mo> <mi> m </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;n&quot;, Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;m&quot;, Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &lt; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <lt /> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "n_", ",", "m_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "z"], ")"]], RowBox[List["k", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k", "-", "1"]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List["z", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], "z"], ")"]], RowBox[List["m", "-", "n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[SuperscriptBox["z", RowBox[List["-", "k"]]], RowBox[List["n", "-", "k"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "1"]], ")"]], "!"]], " ", "z"]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["n", "<", "m"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29