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 JacobiCN

 http://functions.wolfram.com/09.26.06.0010.01

 Input Form

 JacobiCN[z, m] == ((I (-1)^(r + s - 1))/Sqrt[m]) Sum[(j + 1) (((-1)^(k - j) Subscript[dn, k - j][m])/(2 k - 2 j)!) Sum[(((-1)^r Binomial[j, r])/(1 + r)) Subscript[q, r, j] (z - Subscript[z, 0])^(2 k - 1), {r, 0, j}], {k, 0, Infinity}, {j, 0, k}] /; Subscript[z, 0] == 2 r EllipticK[m] + (2 s + 1) I EllipticK[1 - m] && Element[r, Integers] && Element[s, Integers] && Subscript[q, j, 0] == 1 && Subscript[q, j, k] == (1/k) Sum[(j i - k + i) (((-1)^i Subscript[sn, i][m])/ (2 i + 1)!) Subscript[q, j, k - i], {i, 1, k}] && Element[k, Integers] && k > 0 && 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 ) ( - 1 ) r + s - 1 m ( z - z 0 ) k = 0 j = 0 k ( j + 1 ) ( - 1 ) k - j dn k - j ( m ) ( 2 k - 2 j ) ! r = 0 j ( - 1 ) r r + 1 ( j r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["r", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] q r , j ( z - z 0 ) 2 k - 1 /; z 0 2 r K ( m ) + ( 2 s + 1 ) K ( 1 - m ) r TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] s TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] q j , 0 1 q j , k 1 k i = 1 k ( j i + i - k ) ( - 1 ) i sn i ( m ) q j , k - i ( 2 i + 1 ) ! 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]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, 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]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, 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]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sn j ( m ) cn k ( m ) δ KroneckerDelta j + k - n + 1 Condition JacobiCN z m -1 r s -1 m 1 2 z -1 Subscript z 0 -1 j 0 k k 0 j 1 -1 k -1 j Subscript dn k -1 j m 2 k -1 2 j -1 r 0 j -1 r r 1 -1 Binomial j r Subscript q r j z -1 Subscript z 0 2 k -1 Subscript z 0 2 r EllipticK m 2 s 1 EllipticK 1 -1 m r s Subscript q j 0 1 Subscript q j k 1 k -1 i 1 k j i i -1 k -1 i Subscript sn i m Subscript q j k -1 i 2 i 1 -1 k SuperPlus 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

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

 2007-05-02