Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] > Specific values > Specialized values > Case 1F4





http://functions.wolfram.com/07.31.03.0132.01









  


  










Input Form





HypergeometricPFQ[{1}, {n, Subscript[b, 2], Subscript[b, 3], Subscript[b, 4]}, z] == (n - 1)! ((-z)^(1 - n) Product[Pochhammer[1 - Subscript[b, k], n - 1], {k, 2, 4}] HypergeometricPFQ[{}, {Subscript[b, 2] - n + 1, Subscript[b, 3] - n + 1, Subscript[b, 4] - n + 1}, z] - Sum[Product[Pochhammer[1 - Subscript[b, j], k], {j, 2, 4}] (1/((-z)^k (n - k - 1)!)), {k, 1, n - 1}]) /; Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["n", ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"], ",", SubscriptBox["b", "4"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "-", "n"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "4"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "k"]]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["b", "2"], "-", "n", "+", "1"]], ",", RowBox[List[SubscriptBox["b", "3"], "-", "n", "+", "1"]], ",", RowBox[List[SubscriptBox["b", "4"], "-", "n", "+", "1"]]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "4"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "j"]]], ",", "k"]], "]"]]]], ")"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










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> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 4 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mi> n </mi> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> </mrow> <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;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;n&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;3&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&quot;]]], &quot;)&quot;]], RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </mo> <mo> ; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <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;0&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </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> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;j&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <cn type='integer'> 4 </cn> </uplimit> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list /> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <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> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <cn type='integer'> 4 </cn> </uplimit> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </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> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["n_", ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"], ",", SubscriptBox["b_", "4"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "-", "n"]]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "4"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "k"]]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["bb", "2"], "-", "n", "+", "1"]], ",", RowBox[List[SubscriptBox["bb", "3"], "-", "n", "+", "1"]], ",", RowBox[List[SubscriptBox["bb", "4"], "-", "n", "+", "1"]]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "4"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["bb", "j"]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.