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 JacobiP

 http://functions.wolfram.com/05.06.06.0039.01

 Input Form

 JacobiP[n, a, b, z] \[Proportional] JacobiP[n, 0, b, z] + Sum[(1/(b + k + n + 1)) (JacobiP[n, 0, b, z] + (((b + 1 + 2 k) Pochhammer[b + k + 1, n - k])/ ((n - k) Pochhammer[b + k + 1, n - k])) JacobiP[k, 0, b, z]) a, {k, 0, n - 1}] + ((1/(2 n!)) Sum[(Pochhammer[-n, k + 2]/(k + 2)!) Sum[(-1)^(j + k + n) (j + 1) (j + 2) (1 + b + n)^j Pochhammer[3 + b + k, -k + n - 2] StirlingS1[k + 2, j + 2] ((1 + z)/2)^(k + 2), {j, 0, k}], {k, 0, n - 2}]) a^2 + \[Ellipsis] /; (a -> 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "0", ",", "b", ",", "z"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["b", "+", "k", "+", "n", "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "0", ",", "b", ",", "z"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "+", "k", "+", "1"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "+", "k", "+", "1"]], ",", RowBox[List["n", "-", "k"]]]], "]"]]]]], RowBox[List["JacobiP", "[", RowBox[List["k", ",", "0", ",", "b", ",", "z"]], "]"]]]]]], ")"]], "a"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", " "]], RowBox[List["2", RowBox[List["n", "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["k", "+", "2"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "+", "n"]]], " ", RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], RowBox[List["(", RowBox[List["j", "+", "2"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["3", "+", "b", "+", "k"]], ",", RowBox[List[RowBox[List["-", "k"]], "+", "n", "-", "2"]]]], "]"]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["j", "+", "2"]]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], RowBox[List["k", "+", "2"]]]]]]]]]]]]], ")"]], SuperscriptBox["a", "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["a", "\[Rule]", "0"]], ")"]]]]]]

 MathML Form

 P n ( a , b ) ( z ) P n ( 0 , b ) ( z ) + k = 0 n - 1 1 b + k + n + 1 ( P n ( 0 , b ) ( z ) + ( b + 2 k + 1 ) ( b + k + 1 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "k", "+", "1"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ( n - k ) ( b + k + 1 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "k", "+", "1"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] P k ( 0 , b ) ( z ) ) a + 1 2 n ! k = 0 n - 2 ( - n ) k + 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["k", "+", "2"]]], Pochhammer] ( k + 2 ) ! j = 0 k ( - 1 ) j + k + n ( j + 1 ) ( j + 2 ) ( b + n + 1 ) j ( b + k + 3 ) n - k - 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "k", "+", "3"]], ")"]], RowBox[List["n", "-", "k", "-", "2"]]], Pochhammer] S TagBox["S", StirlingS1] k + 2 ( j + 2 ) ( z + 1 2 ) k + 2 a 2 + /; ( a "\[Rule]" 0 ) Condition Proportional JacobiP n a b z JacobiP n 0 b z k 0 n -1 1 b k n 1 -1 JacobiP n 0 b z b 2 k 1 Pochhammer b k 1 n -1 k n -1 k Pochhammer b k 1 n -1 k -1 JacobiP k 0 b z a 1 2 n -1 k 0 n -2 Pochhammer -1 n k 2 k 2 -1 j 0 k -1 j k n j 1 j 2 b n 1 j Pochhammer b k 3 n -1 k -2 StirlingS1 k 2 j 2 z 1 2 -1 k 2 a 2 Rule a 0 [/itex]

 Rule Form

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

 2007-05-02