Generalized hypergeometric function: Summation (formula 07.31.23.0001)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]

Summation

Finite summation

http://functions.wolfram.com/07.31.23.0001.01

Input Form

Sum[(Product[Pochhammer[Subscript[a, j], k], {j, 1, p}]/ Product[Pochhammer[Subscript[b, j], k], {j, 1, q}]) HypergeometricPFQ[{1, (Subscript[a, 1] + k)/n, \[Ellipsis], (Subscript[a, 1] + k + n - 1)/n, \[Ellipsis], (Subscript[a, p] + k)/n, \[Ellipsis], (Subscript[a, p] + k + n - 1)/n}, {(k + 1)/n, \[Ellipsis], (k + n)/n, (Subscript[b, 1] + k)/n, \[Ellipsis], (Subscript[b, 1] + k + n - 1)/n, (Subscript[b, q] + k)/n, \[Ellipsis], (Subscript[b, q] + k + n - 1)/n}, n^(n (p - q - 1)) z^n] (z^k/k!), {k, 0, n - 1}] == HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List[SuperscriptBox["n", RowBox[List["n", RowBox[List["(", RowBox[List["p", "-", "q", "-", "1"]], ")"]]]]], SuperscriptBox["z", "n"]]]]], "]"]], FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]]]]]], "\[Equal]", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> n </mi> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["n", " ", "p"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List[RowBox[List["n", " ", "q"]], "+", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "+", "n", "-", "1"]], "n"], ";", FractionBox[RowBox[List["k", "+", "1"]], "n"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "+", "n", "-", "1"]], "n"], ";", RowBox[List[SuperscriptBox["n", RowBox[List["n", RowBox[List["(", RowBox[List["p", "-", "q", "-", "1"]], ")"]]]]], SuperscriptBox["z", "n"]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> n </mi> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["n", " ", "p"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List[RowBox[List["n", " ", "q"]], "+", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "+", "n", "-", "1"]], "n"], ";", FractionBox[RowBox[List["k", "+", "1"]], "n"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "+", "n", "-", "1"]], "n"], ";", RowBox[List[SuperscriptBox["n", RowBox[List["n", RowBox[List["(", RowBox[List["p", "-", "q", "-", "1"]], ")"]]]]], SuperscriptBox["z", "n"]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </annotation-xml> </semantics> </math>

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]