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 Fibonacci

 http://functions.wolfram.com/05.12.23.0001.01

 Input Form

 Sum[KroneckerDelta[n - Sum[Subscript[k, j], {j, 1, p}]]*Product[Fibonacci[Subscript[k, j] + 1, z], {j, 1, p}], {Subscript[k, 1], 1, n}, {Subscript[k, 2], 1, n}, â€¦, {Subscript[k, p], 1, n}] == Sum[Binomial[-j + p + n - 1, j]* Binomial[-2*j + p + n - 1, p - 1]*z^(n - 2*j), {j, 0, Floor[n/2]}] /; n âˆˆ Integers && n >= 0 && p âˆˆ Integers && p >= 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "1"]], "n"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "p"], "=", "1"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Fibonacci", "[", RowBox[List[RowBox[List[SubscriptBox["k", "j"], "+", "1"]], ",", "z"]], "]"]]]]]]]]]]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["\[LeftFloor]", RowBox[List["n", "/", "2"]], "\[RightFloor]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "p", "-", "1", "-", "j"]], ",", "j"]], "]"]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "p", "-", "1", "-", RowBox[List["2", "j"]]]], ",", RowBox[List["p", "-", "1"]]]], "]"]], SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", "j"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "1"]]]]]]]]

 MathML Form

 k 1 = 1 n k 2 = 1 n k p = 1 n δ KroneckerDelta n - j = 1 p k j j = 1 p F TagBox["F", Fibonacci] k j + 1 ( z ) j = 0 n / 2 ( p - j + n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["p", "-", "j", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( p - 2 j + n - 1 p - 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["p", "-", RowBox[List["2", " ", "j"]], "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["p", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] z n - 2 j /; n p + Condition Subscript k 2 1 n Subscript k 1 1 n Subscript k p 1 n KroneckerDelta n -1 j 1 p Subscript k j j 1 p Fibonacci Subscript k j 1 z j 0 n 2 -1 Binomial p -1 j n -1 j Binomial p -1 2 j n -1 p -1 z n -1 2 j n p SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "1"], "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "2"], "=", "1"]], "n_"], RowBox[List["\[Ellipsis]_", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "p_"], "=", "1"]], "n_"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["n_", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j_", "=", "1"]], "p_"], SubscriptBox["k_", "j_"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j_", "=", "1"]], "p_"], RowBox[List["Fibonacci", "[", RowBox[List[RowBox[List[SubscriptBox["k_", "j_"], "+", "1"]], ",", "z_"]], "]"]]]]]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "p", "-", "1", "-", "j"]], ",", "j"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "p", "-", "1", "-", RowBox[List["2", " ", "j"]]]], ",", RowBox[List["p", "-", "1"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", " ", "j"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]]]]]]]]]]

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

 2002-12-18