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 JacobiSD

 http://functions.wolfram.com/09.35.06.0006.01

 Input Form

 JacobiSD[z, m] == Sum[(j + 1) (((-1)^(k - j) Subscript[sn, k - j][m])/ (2 k - 2 j + 1)!) Sum[(((-1)^r Binomial[j, r])/(1 + r)) Subscript[q, r, j] z^(2 k + 1), {r, 0, j}], {k, 0, Infinity}, {j, 0, k}] /; Subscript[q, j, 0] == 1 && Subscript[q, j, k] == (1/k) Sum[(j i - k + i) (((-1)^i Subscript[dn, i][m])/(2 i)!) 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

 sd ( z m ) k = 0 j = 0 k ( j + 1 ) ( - 1 ) k - j sn k - j ( m ) ( 2 k - 2 j + 1 ) ! 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 2 k + 1 /; q j , 0 1 q j , k 1 k i = 1 k ( j i + i - k ) ( - 1 ) i dn i ( m ) q j , k - i ( 2 i ) ! 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 JacobiSD z m j 0 k k 0 j 1 -1 k -1 j Subscript sn k -1 j m 2 k -1 2 j 1 -1 r 0 j -1 r r 1 -1 Binomial j r Subscript q r j z 2 k 1 Subscript q j 0 1 Subscript q j k 1 k -1 i 1 k j i i -1 k -1 i Subscript dn i m Subscript q j k -1 i 2 i -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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List[SubscriptBox["sn", RowBox[List["k", "-", "j"]]], "[", "m", "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], ")"]], " ", SubscriptBox["q", RowBox[List["r", ",", "j"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], RowBox[List["1", "+", "r"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["q", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["q", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "i"]], "-", "k", "+", "i"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", RowBox[List[SubscriptBox["dn", "i"], "[", "m", "]"]]]], ")"]], " ", SubscriptBox["q", RowBox[List["j", ",", RowBox[List["k", "-", "i"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "i"]], ")"]], "!"]]]]], "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]], "&&", 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