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 ZernikeR

 http://functions.wolfram.com/05.18.20.0005.01

 Input Form

 D[ZernikeR[n, m, z], {z, h}] == Piecewise[{{(((-1)^((n - m)/2) Gamma[(2 + m + n)/2])/ (Gamma[(2 - m + n)/2] m!)) Sum[(Binomial[h, k] Pochhammer[m - h + k + 1, h - k] z^(m - h) Sum[(Pochhammer[2 j - k + 1, 2 (k - j)]/(k - j)!) (2 z)^(2 j) ((Pochhammer[(2 + m + n)/2, j] Pochhammer[(m - n)/2, j])/ Pochhammer[1 + m, j]) Hypergeometric2F1[(2 + m + n)/2 + j, (m - n)/2 + j, 1 + m + j, z^2], {j, 0, k}])/2^k, {k, 0, h}], Element[(n - m)/2, Integers]}}, 0] /; Element[h, Integers] && h >= 0 && Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && m <= n

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "h"]], "}"]]], RowBox[List["(", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]], ")"]]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["n", "-", "m"]], "2"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["2", "+", "m", "+", "n"]], "2"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["2", "-", "m", "+", "n"]], "2"], "]"]], RowBox[List["m", "!"]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "h"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["h", ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "h", "+", "k", "+", "1"]], ",", RowBox[List["h", "-", "k"]]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "k"]]], SuperscriptBox["z", RowBox[List["m", "-", "h"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", "j"]], "-", "k", " ", "+", "1"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]]]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "j"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "m", "+", "n"]], "2"], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], ",", "j"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "j"]], "]"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", "+", "m", "+", "n"]], "2"], "+", "j"]], ",", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], "+", "j"]], ",", RowBox[List["1", "+", "m", "+", "j"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]]]]]]]], ",", RowBox[List[FractionBox[RowBox[List["n", "-", "m"]], "2"], "\[Element]", "Integers"]]]], "}"]], "}"]], ",", "0"]], "]"]]]], "/;", RowBox[List[RowBox[List["h", "\[Element]", "Integers"]], "\[And]", RowBox[List["h", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]

 MathML Form

 h R n m ( z ) z h ( - 1 ) n - m 2 Γ ( m + n + 2 2 ) Γ ( 2 - m + n 2 ) m ! k = 0 h ( h k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["h", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( k + m - h + 1 ) h - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "+", "m", "-", "h", "+", "1"]], ")"]], RowBox[List["h", "-", "k"]]], Pochhammer] 2 - k z m - h j = 0 k ( 2 j - k + 1 ) 2 ( k - j ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k", "+", "1"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]]]]], Pochhammer] ( m + n + 2 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["m", "+", "n", "+", "2"]], "2"], ")"]], "j"], Pochhammer] ( m - n 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["m", "-", "n"]], "2"], ")"]], "j"], Pochhammer] ( k - j ) ! ( m + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "j"], Pochhammer] ( 2 z ) 2 j 2 F 1 ( j + m + n + 2 2 , j + m - n 2 ; j + m + 1 ; z 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["m", "+", "n", "+", "2"]], "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["m", "-", "n"]], "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "+", "m", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] n - m 2 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] /; h n m m n Condition z h ZernikeR n m z Piecewise -1 n -1 m 2 -1 Gamma m n 2 2 -1 Gamma 2 -1 m n 2 -1 m -1 k 0 h Binomial h k Pochhammer k m -1 h 1 h -1 k 2 -1 k z m -1 h j 0 k Pochhammer 2 j -1 k 1 2 k -1 j Pochhammer m n 2 2 -1 j Pochhammer m -1 n 2 -1 j k -1 j Pochhammer m 1 j -1 2 z 2 j HypergeometricPFQ j m n 2 2 -1 j m -1 n 2 -1 j m 1 z 2 n -1 m 2 -1 h n m m n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "h_"]], "}"]]]]], RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["n", "-", "m"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "h"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["h", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "h", "+", "k", "+", "1"]], ",", RowBox[List["h", "-", "k"]]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List["m", "-", "h"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "-", "k", "+", "1"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["2", " ", "j"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], "+", "j"]], ",", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], "+", "j"]], ",", RowBox[List["1", "+", "m", "+", "j"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "j"]], "]"]]]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "m", "+", "n"]], ")"]]]], "]"]], " ", RowBox[List["m", "!"]]]]], RowBox[List[FractionBox[RowBox[List["n", "-", "m"]], "2"], "\[Element]", "Integers"]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[RowBox[List["h", "\[Element]", "Integers"]], "&&", RowBox[List["h", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]]]

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

 2007-05-02