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 StruveL

 http://functions.wolfram.com/03.10.17.0016.01

 Input Form

 StruveL[\[Nu], z] == (2^(n - 1) Pochhammer[1 - \[Nu], n - 1] (2 (n - \[Nu]) Sum[(((-1)^k (n - k)!)/(k! (n - 2 k)! Pochhammer[1 - \[Nu], k] Pochhammer[\[Nu] - n, k])) (z^2/4)^k StruveL[\[Nu] - n, z], {k, 0, Floor[n/2]}] + z Sum[(((-1)^k (n - k - 1)!)/(k! (n - 2 k - 1)! Pochhammer[1 - \[Nu], k] Pochhammer[1 + \[Nu] - n, k])) (z^2/4)^k StruveL[\[Nu] - n - 1, z], {k, 0, Floor[(n - 1)/2]}]))/z^n - (1/Sqrt[Pi]) (z/2)^(\[Nu] - 1) Sum[(Pochhammer[1 - \[Nu], j]/(Gamma[-j + \[Nu] + 1/2] (z^2/4)^j)) Sum[(((-1)^k (j - k)!)/(k! (j - 2 k)! Pochhammer[1 - \[Nu], k] Pochhammer[\[Nu] - j, k])) (z^2/4)^k, {k, 0, Floor[j/2]}], {j, 0, n - 1}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveL", "[", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]]]]]], "+", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"], RowBox[List["StruveL", "[", 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["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "j"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["j", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "-", "k"]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "j"]], ",", "k"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 L StruveL ν ( z ) 2 n - 1 z - n ( 1 - ν ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( 2 ( n - ν ) k = 0 n 2 ( - 1 ) k ( n - k ) ! k ! ( n - 2 k ) ! ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "n"]], ")"]], "k"], Pochhammer] ( z 2 4 ) k L StruveL ν - n ( z ) + z k = 0 n - 1 2 ( - 1 ) k ( n - k - 1 ) ! k ! ( n - 2 k - 1 ) ! ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν - n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "n", "+", "1"]], ")"]], "k"], Pochhammer] ( z 2 4 ) k L StruveL 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 k = 0 j 2 ( - 1 ) k ( j - k ) ! ( z 2 4 ) k k ! ( j - 2 k ) ! ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "j"]], ")"]], "k"], Pochhammer] /; n Condition StruveL ν z 2 n -1 z -1 n Pochhammer 1 -1 ν n -1 2 n -1 ν k 0 n 2 -1 -1 k n -1 k k n -1 2 k Pochhammer 1 -1 ν k Pochhammer ν -1 n k -1 z 2 4 -1 k StruveL ν -1 n z z k 0 n -1 2 -1 -1 k n -1 k -1 k n -1 2 k -1 Pochhammer 1 -1 ν k Pochhammer ν -1 n 1 k -1 z 2 4 -1 k StruveL ν -1 n -1 z -1 1 1 2 -1 z 2 -1 ν -1 j 0 n -1 Pochhammer 1 -1 ν j Gamma ν -1 j 1 2 z 2 4 -1 j -1 k 0 j 2 -1 -1 k j -1 k z 2 4 -1 k k j -1 2 k Pochhammer 1 -1 ν k Pochhammer ν -1 j k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveL", "[", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "n"]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], "-", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["j", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", "k"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "j"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "j"]]]]]]]], 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