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 KelvinBer

 http://functions.wolfram.com/03.14.06.0017.01

 Input Form

 KelvinBer[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (1/2) Sum[z^(4 k)/(2^(4 k) (k!^2 (2 k)!)), {k, 0, n}] + (1/2) Sum[((-1)^k Pochhammer[1/4, k] Pochhammer[3/4, k] z^(4 k))/(16^k (Pochhammer[1/2, k]^3 k!^3)), {k, 0, n}] == KelvinBer[z]^2 - ((2^(-5 - 4 n) z^(4 (1 + n)))/(Gamma[2 + n]^2 Gamma[3 + 2 n])) HypergeometricPFQ[{1}, {3/2 + n, 2 + n, 2 + n, 2 + n}, z^4/64] + (((-1)^n z^(4 (1 + n)) Gamma[5/2 + 2 n])/ (2 Sqrt[Pi] Gamma[3 + 2 n]^3)) HypergeometricPFQ[{1, 5/4 + n, 7/4 + n}, {3/2 + n, 3/2 + n, 3/2 + n, 2 + n, 2 + n, 2 + n}, -(z^4/16)] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["KelvinBer", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["16", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "4"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "4"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "3"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "3"]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["KelvinBer", "[", "z", "]"]], "2"], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["4", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " "]], RowBox[List[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "n"]], "]"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "+", RowBox[List["2", " ", "n"]]]], "]"]], " "]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], "]"]], "3"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["5", "4"], "+", "n"]], ",", RowBox[List[FractionBox["7", "4"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "16"]]]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 ber ( z ) 2 F ( z ) /; ( ( F n ( z ) 1 2 k = 0 n 16 - k ( - 1 ) k ( 1 4 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "4"], ")"]], "k"], Pochhammer] ( 3 4 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "4"], ")"]], "k"], Pochhammer] z 4 k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 3 ( k ! ) 3 + 1 2 k = 0 n 2 - 4 k z 4 k ( k ! ) 2 ( 2 k ) ! ber ( z ) 2 - 2 - 4 n - 5 z 4 ( n + 1 ) Γ ( n + 2 ) 2 Γ ( 2 n + 3 ) 1 F 4 ( 1 ; n + 3 2 , n + 2 , n + 2 , n + 2 ; z 4 64 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "4"], "64"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] + ( - 1 ) n z 4 ( n + 1 ) Γ ( 2 n + 5 2 ) 2 π Γ ( 2 n + 3 ) 3 3 F 6 ( 1 , n + 5 4 , n + 7 4 ; n + 3 2 , n + 3 2 , n + 3 2 , n + 2 , n + 2 , n + 2 ; - z 4 16 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "6"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["7", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "16"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition KelvinBer z 2 Subscript F z Subscript F n z 1 2 k 0 n 16 -1 k -1 k Pochhammer 1 4 k Pochhammer 3 4 k z 4 k Pochhammer 1 2 k 3 k 3 -1 1 2 k 0 n 2 -4 k z 4 k k 2 2 k -1 KelvinBer z 2 -1 2 -4 n -5 z 4 n 1 Gamma n 2 2 Gamma 2 n 3 -1 HypergeometricPFQ 1 n 3 2 n 2 n 2 n 2 z 4 64 -1 -1 n z 4 n 1 Gamma 2 n 5 2 2 1 2 Gamma 2 n 3 3 -1 HypergeometricPFQ 1 n 5 4 n 7 4 n 3 2 n 3 2 n 3 2 n 2 n 2 n 2 -1 z 4 16 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["KelvinBer", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["16", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "4"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "4"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "3"]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["KelvinBer", "[", "z", "]"]], "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["4", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "n"]], "]"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "+", RowBox[List["2", " ", "n"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["5", "4"], "+", "n"]], ",", RowBox[List[FractionBox["7", "4"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "16"]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], "]"]], "3"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02