Generalized hypergeometric function 1F2: Specific values (formula 07.22.03.0312)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁},{b₁,b₂},z]

Specific values

Specialized values

For fixed z and with symbolical integers in parameters

For fixed z and a₁=m, b₁=1/2+n, b₂=1/2+p

http://functions.wolfram.com/07.22.03.0312.01

Input Form

HypergeometricPFQ[{1 + j}, {1/2 + m, 1/2 + n}, z] == (((-1)^j Pochhammer[1/2, m] Pochhammer[1/2, n])/(4 j!)) Sum[((-1)^k/(m - k)!) Sum[(((-1)^i z^(-i - 1))/(k - i)!) (((-1 - 2 i + 2 m) (1 - 2 i + 2 m) i!)/((i - j)! (1 - 2 i + 2 k) Pochhammer[-(1/2), -i + n]) + Sum[(-1)^p Binomial[j, p] Pochhammer[-(1/2) - j + n, j - p] ((1 + 12 i^2 - 2 n + 4 m (-2 + m + 2 n - 2 p) + 2 p + 2 i (5 - 8 m - 4 n + 4 p) + 4 z)/Pochhammer[1/2, -i + n - p] + 4 Pi z^((i - n + p)/2) ((-(i - m)) ((1 + i - m) (1 + i - n + p) + 2 z) StruveL[-1 - i + n - p, 2 Sqrt[z]] + Sqrt[z] ((i - m) (1 + i - m) + z) StruveL[-i + n - p, 2 Sqrt[z]])), {p, 0, j}]), {i, 0, k}], {k, 0, m}] /; Element[j, Integers] && j >= 0 && Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "j"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "+", "n"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]]], RowBox[List["4", RowBox[List["j", "!"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "i"]], "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i"]], ")"]], "!"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["i", "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "n"]]]], "]"]]]]], "+", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "p"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j", "+", "n"]], ",", RowBox[List["j", "-", "p"]]]], "]"]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["12", " ", SuperscriptBox["i", "2"]]], "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["4", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m", "+", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "p"]]]], ")"]]]], "+", RowBox[List["2", " ", "p"]], "+", RowBox[List["2", " ", "i", " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["8", " ", "m"]], "-", RowBox[List["4", " ", "n"]], "+", RowBox[List["4", " ", "p"]]]], ")"]]]], "+", RowBox[List["4", " ", "z"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "i"]], "+", "n", "-", "p"]]]], "]"]]], "+", RowBox[List["4", " ", "\[Pi]", " ", SuperscriptBox["z", FractionBox[RowBox[List["i", "-", "n", "+", "p"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["i", "-", "m"]], ")"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "i", "-", "n", "+", "p"]], ")"]]]], "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "i", "+", "n", "-", "p"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]]]], "+", "z"]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "i"]], "+", "n", "-", "p"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> i </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> i </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> i </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["n", "-", "i"]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> j </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> <mtr> <mtd> <mi> p </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["p", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> p </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n", "-", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "p"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mfrac> <mrow> <mi> i </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> p </mi> </mrow> <mn> 2 </mn> </mfrac> <mtext> </mtext> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msup> <mi> i </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["-", "i"]], "+", "n", "-", "p"]]], Pochhammer] </annotation> </semantics> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> j </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <ci> i </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <ci> Binomial </ci> <ci> j </ci> <ci> p </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <ci> StruveL </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> StruveL </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -8 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> p </ci> </apply> <cn type='integer'> 5 </cn> </apply> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> j </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "j_"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "m_"]], ",", RowBox[List[FractionBox["1", "2"], "+", "n_"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "i"]], "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["i", "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "i"]], "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "n"]]]], "]"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "p"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j", "+", "n"]], ",", RowBox[List["j", "-", "p"]]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["12", " ", SuperscriptBox["i", "2"]]], "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["4", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m", "+", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "p"]]]], ")"]]]], "+", RowBox[List["2", " ", "p"]], "+", RowBox[List["2", " ", "i", " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["8", " ", "m"]], "-", RowBox[List["4", " ", "n"]], "+", RowBox[List["4", " ", "p"]]]], ")"]]]], "+", RowBox[List["4", " ", "z"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "i"]], "+", "n", "-", "p"]]]], "]"]]], "+", RowBox[List["4", " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["i", "-", "n", "+", "p"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["i", "-", "m"]], ")"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "i", "-", "n", "+", "p"]], ")"]]]], "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "i", "+", "n", "-", "p"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]]]], "+", "z"]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "i"]], "+", "n", "-", "p"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i"]], ")"]], "!"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], "!"]]]]]]], RowBox[List["4", " ", RowBox[List["j", "!"]]]]], "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

Contributed by

Brychkov Yu.A. (2006)

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

2007-05-02

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