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 StruveH

 http://functions.wolfram.com/03.09.20.0020.01

 Input Form

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

 Standard Form

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")"]], "j"], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n H StruveH ν ( z ) z n z - n m = 0 n ( - 1 ) m + n ( n m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["m", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ν ) n - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["n", "-", "m"]]], Pochhammer] k = 0 m ( - 1 ) k - 1 2 2 k - m ( - m ) 2 ( m - k ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]], Pochhammer] ( ν ) k TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "k"], Pochhammer] ( m - k ) ! ( z 2 j = 0 k - 1 ( k - j - 1 ) ! ( z 2 4 ) j j ! ( k - 2 j - 1 ) ! ( 1 - k - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν ) j + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] H StruveH ν - 1 ( z ) - j = 0 k ( k - j ) ! ( z 2 4 ) j j ! ( k - 2 j ) ! ( 1 - k - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν ) j TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "j"], Pochhammer] H StruveH ν ( z ) ) + 2 - ν z - n + ν + 1 π Γ ( ν + 1 2 ) i = 1 n - 1 m = 0 i ( - 1 ) i + m ( i m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["m", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ν ) i - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["i", "-", "m"]]], Pochhammer] k = 0 m ( - 1 ) k - 1 2 2 k - m ( - m ) 2 ( m - k ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]], Pochhammer] ( ν ) k TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "k"], Pochhammer] ( m - k ) ! j = 0 k - 1 2 - 2 j ( k - j - 1 ) ! ( 2 j - n + ν + 2 ) n - i - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", "n", "+", "\[Nu]", "+", "2"]], ")"]], RowBox[List["n", "-", "i", "-", "1"]]], Pochhammer] z 2 j j ! ( k - 2 j - 1 ) ! ( 1 - k - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν ) j + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z n StruveH ν z z -1 n m 0 n -1 m n Binomial n m Pochhammer -1 ν n -1 m k 0 m -1 k -1 2 2 k -1 m Pochhammer -1 m 2 m -1 k Pochhammer ν k m -1 k -1 z 2 -1 j 0 k -1 k -1 j -1 z 2 4 -1 j j k -1 2 j -1 Pochhammer 1 -1 k -1 ν j Pochhammer ν j 1 -1 StruveH ν -1 z -1 j 0 k k -1 j z 2 4 -1 j j k -1 2 j Pochhammer 1 -1 k -1 ν j Pochhammer ν j -1 StruveH ν z 2 -1 ν z -1 n ν 1 1 2 Gamma ν 1 2 -1 m 0 i i 1 n -1 -1 i m Binomial i m Pochhammer -1 ν i -1 m k 0 m -1 k -1 2 2 k -1 m Pochhammer -1 m 2 m -1 k Pochhammer ν k m -1 k -1 j 0 k -1 2 -2 j k -1 j -1 Pochhammer 2 j -1 n ν 2 n -1 i -1 z 2 j j k -1 2 j -1 Pochhammer 1 -1 k -1 ν j Pochhammer ν j 1 -1 n [/itex]

 Rule Form

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

 2007-05-02