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 StruveH

 http://functions.wolfram.com/03.09.17.0013.01

 Input Form

 StruveH[\[Nu], z] == (2^(n - 1) Pochhammer[1 + \[Nu], n - 1] (2 (n + \[Nu]) HypergeometricPFQ[{1, (1 - n)/2, -(n/2)}, {1, -n, -n - \[Nu], 1 + \[Nu]}, -z^2] StruveH[n + \[Nu], z] - z HypergeometricPFQ[{1, (1 - n)/2, 1 - n/2}, {1, 1 - n, 1 - n - \[Nu], 1 + \[Nu]}, -z^2] StruveH[1 + n + \[Nu], z]))/z^n + ((2^(-1 - \[Nu]) z^(1 + \[Nu]))/Sqrt[Pi]) Sum[(Pochhammer[1 + \[Nu], j]/Gamma[5/2 + j + \[Nu]]) HypergeometricPFQ[{1, (1 - j)/2, -(j/2)}, {1, -j, -j - \[Nu], 1 + \[Nu]}, -z^2], {j, 0, n - 1}] /; 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[SuperscriptBox["2", RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["n", "-", "1"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "-", RowBox[List["z", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "n", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Nu]"]]]]], SqrtBox["\[Pi]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "+", "j", "+", "\[Nu]"]], "]"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "j"]], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 H StruveH ν ( z ) 2 n - 1 z - n ( ν + 1 ) n - 1 TagBox[SubscriptBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], "(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( 2 ( n + ν ) 3 F 4 ( 1 , 1 - n 2 , - n 2 ; 1 , - n , - n - ν , ν + 1 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", 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["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], 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] H StruveH n + ν ( z ) - z 3 F 4 ( 1 , 1 - n 2 , 1 - n 2 ; 1 , 1 - n , 1 - n - ν , ν + 1 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "n", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], 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] H StruveH n + ν + 1 ( z ) ) + 2 - ν - 1 z ν + 1 π j = 0 n - 1 ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] Γ ( j + ν + 5 2 ) 3 F 4 ( 1 , 1 - j 2 , - j 2 ; 1 , - j , - j - ν , ν + 1 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", 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["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], 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 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition StruveH ν z 2 n -1 Pochhammer Subscript z -1 n ν 1 n -1 2 n ν HypergeometricPFQ 1 1 -1 n 2 -1 -1 n 2 -1 1 -1 n -1 n -1 ν ν 1 -1 z 2 StruveH n ν z -1 z HypergeometricPFQ 1 1 -1 n 2 -1 1 -1 n 2 -1 1 1 -1 n 1 -1 n -1 ν ν 1 -1 z 2 StruveH n ν 1 z 2 -1 ν -1 z ν 1 1 2 -1 j 0 n -1 Pochhammer ν 1 j Gamma j ν 5 2 -1 HypergeometricPFQ 1 1 -1 j 2 -1 -1 j 2 -1 1 -1 j -1 j -1 ν ν 1 -1 z 2 n [/itex]

 Rule Form

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

 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02