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 StruveH

 http://functions.wolfram.com/03.09.17.0019.01

 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]) (z/2)^(\[Nu] - 1) Sum[(Pochhammer[1 - \[Nu], j]/(Gamma[-j + \[Nu] + 1/2] (-(z^2/4))^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", SqrtBox["\[Pi]"]], 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[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "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[RowBox[List["(", RowBox[List["-", "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 ν - n ( z ) - 𝒞 n - 1 ( ν , z ) H StruveH ν - n - 1 ( z ) + 1 π ( z 2 ) ν - 1 j = 0 n - 1 ( 1 - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] Γ ( ν - j + 1 2 ) ( - z 2 4 ) j 2 F 3 ( 1 - j 2 , - j 2 ; 1 - ν , - j , ν - j ; - 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["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "j"]], 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 ( 1 - ν ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "n"], Pochhammer] 2 F 3 ( 1 - n 2 , - n 2 ; 1 - ν , - n , ν - n ; - 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["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "n"]], 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 Condition StruveH ν z Subscript 𝒞 n ν z StruveH ν -1 n z -1 Subscript 𝒞 n -1 ν z StruveH ν -1 n -1 z 1 1 2 -1 z 2 -1 ν -1 j 0 n -1 Pochhammer 1 -1 ν j Gamma ν -1 j 1 2 -1 z 2 4 -1 j -1 HypergeometricPFQ 1 -1 j 2 -1 -1 j 2 -1 1 -1 ν -1 j ν -1 j -1 z 2 Subscript 𝒞 n ν z -2 n z -1 n Pochhammer 1 -1 ν n HypergeometricPFQ 1 -1 n 2 -1 -1 n 2 -1 1 -1 ν -1 n ν -1 n -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[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "j"]]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "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