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 StruveH

 http://functions.wolfram.com/03.09.17.0015.01

 Input Form

 StruveH[\[Nu], z] == Subscript[\[ScriptCapitalC], n][\[Nu], z] StruveH[\[Nu] + n, z] - Subscript[\[ScriptCapitalC], n - 1][\[Nu], z] StruveH[\[Nu] + n + 1, z] + (1/(Sqrt[Pi] Gamma[5/2 + \[Nu]])) (z/2)^(\[Nu] + 1) Sum[(Pochhammer[1 + \[Nu], j]/Pochhammer[5/2 + \[Nu], j]) HypergeometricPFQ[{(1 - j)/2, -(j/2)}, {1 + \[Nu], -j, -j - \[Nu]}, -z^2], {j, 0, n - 1}] /; Subscript[\[ScriptCapitalC], n][\[Nu], z] == (2^n Pochhammer[1 + \[Nu], n] HypergeometricPFQ[{(1 - n)/2, -(n/2)}, {1 + \[Nu], -n, -n - \[Nu]}, -z^2])/z^n && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n"]], ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n", "+", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["5", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "j"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "n"]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 H StruveH ν ( z ) 𝒞 n ( ν , z ) H StruveH n + ν ( z ) - 𝒞 n - 1 ( ν , z ) H StruveH n + ν + 1 ( z ) + 1 π Γ ( ν + 5 2 ) ( z 2 ) ν + 1 j = 0 n - 1 ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] ( ν + 5 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["5", "2"]]], ")"]], "j"], Pochhammer] 2 F 3 ( 1 - j 2 , - j 2 ; ν + 1 , - j , - j - ν ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "j"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["j", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; 𝒞 n ( ν , z ) 2 n z - n ( ν + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "n"], Pochhammer] 2 F 3 ( 1 - n 2 , - n 2 ; ν + 1 , - n , - n - ν ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] n + Condition StruveH ν z Subscript 𝒞 n ν z StruveH n ν z -1 Subscript 𝒞 n -1 ν z StruveH n ν 1 z 1 1 2 Gamma ν 5 2 -1 z 2 -1 ν 1 j 0 n -1 Pochhammer ν 1 j Pochhammer ν 5 2 j -1 HypergeometricPFQ 1 -1 j 2 -1 -1 j 2 -1 ν 1 -1 j -1 j -1 ν -1 z 2 Subscript 𝒞 n ν z 2 n z -1 n Pochhammer ν 1 n HypergeometricPFQ 1 -1 n 2 -1 -1 n 2 -1 ν 1 -1 n -1 n -1 ν -1 z 2 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n"]], ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n", "+", "1"]], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "j"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["5", "2"], "+", "\[Nu]"]], ",", "j"]], "]"]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "n"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02