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 JacobiCN

 http://functions.wolfram.com/09.26.06.0007.01

 Input Form

 JacobiCN[z, m] == Sum[((-1)^k Subscript[cn, k][m] z^(2 k))/(2 k)!, {k, 0, Infinity}] /; Subscript[sn, 0][m] == 1 && Subscript[sn, n][m] == Sum[Binomial[2 n, 2 j] Subscript[cn, j][m] Subscript[dn, k][m] KroneckerDelta[j + k - n], {j, 0, n}, {k, 0, n}] && Subscript[cn, 0][m] == 1 && Subscript[cn, n][m] == Sum[Binomial[2 n - 1, 2 j + 1] Subscript[sn, j][m] Subscript[dn, k][m] KroneckerDelta[j + k - n + 1], {j, 0, n - 1}, {k, 0, n - 1}] && Subscript[dn, 0][m] == 1 && Subscript[dn, n][m] == m Sum[Binomial[2 n - 1, 2 j + 1] Subscript[sn, j][m] Subscript[cn, k][m] KroneckerDelta[j + k - n + 1], {j, 0, n - 1}, {k, 0, n - 1}]

 Standard Form

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 MathML Form

 cn ( z m ) k = 0 ( - 1 ) k cn k ( m ) z 2 k ( 2 k ) ! /; sn 0 ( m ) 1 sn n ( m ) j = 0 n k = 0 n ( 2 n 2 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["2", " ", "n"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] cn j ( m ) dn k ( m ) δ KroneckerDelta j + k - n cn 0 ( m ) 1 cn n ( m ) j = 0 n - 1 k = 0 n - 1 ( 2 n - 1 2 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] sn j ( m ) dn k ( m ) δ KroneckerDelta j + k - n + 1 dn 0 ( m ) 1 dn n ( m ) m j = 0 n - 1 k = 0 n - 1 ( 2 n - 1 2 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] sn j ( m ) cn k ( m ) δ KroneckerDelta j + k - n + 1 Condition JacobiCN z m k 0 -1 k Subscript cn k m z 2 k 2 k -1 Subscript sn 0 m 1 Subscript sn n m k 0 n j 0 n Binomial 2 n 2 j Subscript cn j m Subscript dn k m KroneckerDelta j k -1 n Subscript cn 0 m 1 Subscript cn n m k 0 n -1 j 0 n -1 Binomial 2 n -1 2 j 1 Subscript sn j m Subscript dn k m KroneckerDelta j k -1 n 1 Subscript dn 0 m 1 Subscript dn n m m k 0 n -1 j 0 n -1 Binomial 2 n -1 2 j 1 Subscript sn j m Subscript cn k m KroneckerDelta j k -1 n 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SubscriptBox["cn", "k"], "[", "m", "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["sn", "0"], "[", "m", "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["sn", "n"], "[", "m", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", RowBox[List[SubscriptBox["cn", "j"], "[", "m", "]"]], " ", RowBox[List[SubscriptBox["dn", "k"], "[", "m", "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["j", "+", "k", "-", "n"]], "]"]]]]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["cn", "0"], "[", "m", "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["cn", "n"], "[", "m", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List[SubscriptBox["sn", "j"], "[", "m", "]"]], " ", RowBox[List[SubscriptBox["dn", "k"], "[", "m", "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["j", "+", "k", "-", "n", "+", "1"]], "]"]]]]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["dn", "0"], "[", "m", "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["dn", "n"], "[", "m", "]"]], "\[Equal]", RowBox[List["m", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List[SubscriptBox["sn", "j"], "[", "m", "]"]], " ", RowBox[List[SubscriptBox["cn", "k"], "[", "m", "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["j", "+", "k", "-", "n", "+", "1"]], "]"]]]]]]]]]]]]]]]]]]]]

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

 2007-05-02