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variants of this functions
JacobiP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Identities > Functional identities > Relations between contiguous functions > Normalized recurrence relation





http://functions.wolfram.com/07.15.17.0007.01









  


  










Input Form





z p[\[Nu], z] == p[\[Nu] + 1, z] + ((b^2 - a^2)/((a + b + 2 \[Nu]) (a + b + 2 \[Nu] + 2))) p[\[Nu], z] + ((4 \[Nu] (a + \[Nu]) (b + \[Nu]) (a + b + \[Nu]))/ ((a + b + 2 \[Nu] - 1) (a + b + 2 \[Nu])^2 (a + b + 2 \[Nu] + 1))) p[\[Nu] - 1, z] /; p[\[Nu], z] == ((2^\[Nu] Gamma[\[Nu] + 1])/Pochhammer[a + b + \[Nu] + 1, \[Nu]]) JacobiP[\[Nu], a, b, z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["a", "2"]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], " "]]], RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["4", "\[Nu]", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " "]]], RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]], RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["z_", " ", RowBox[List["p", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["a", "2"]]], ")"]], " ", RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]]]]]]]]]]










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





2001-10-29





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