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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.06.0062.01

 Input Form

 Hypergeometric2F1[a, a + n, a + m, z] == (((Gamma[a + m] Gamma[n])/(Gamma[m] Gamma[a + n])) Sum[(Pochhammer[a, k] Pochhammer[1 - m, k])/(k! Pochhammer[1 - n, k])/ z^k, {k, 0, m - 1}])/(-z)^a + ((Pochhammer[a, m] (n - m)!)/n!) (-1)^m (-z)^(-a - n) Sum[(Pochhammer[a + n, k] Pochhammer[1 - m + n, k])/ (k! Pochhammer[1 + n, k])/z^k, {k, 0, Infinity}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && m <= n && !IntervalMemberQ[Interval[{0, 1}], z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["a", "+", "n"]], ",", RowBox[List["a", "+", "m"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "m"]], "]"]], RowBox[List["Gamma", "[", "n", "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "m", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]], RowBox[List["n", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "n"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 2 F 1 ( a , a + n ; a + m ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "m"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] Γ ( a + m ) Γ ( n ) ( - z ) - a Γ ( m ) Γ ( a + n ) k = 0 m - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( 1 - m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"], Pochhammer] z - k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] + ( - 1 ) m ( a ) m TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "m"], Pochhammer] ( n - m ) ! ( - z ) - a - n n ! k = 0 ( a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "k"], Pochhammer] ( - m + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "k"], Pochhammer] z - k /; n m m n z ( 0 , 1 ) 2 F 1 ( a , a + n ; a + m ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "m"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] Γ ( a + m ) Γ ( n ) ( - z ) - a Γ ( m ) Γ ( a + n ) k = 0 m - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( 1 - m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"], Pochhammer] z - k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] + ( - 1 ) m ( a ) m TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "m"], Pochhammer] ( n - m ) ! ( - z ) - a - n n ! k = 0 ( a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "k"], Pochhammer] ( - m + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "k"], Pochhammer] z - k /; n m m n z ( 0 , 1 ) [/itex]

 Rule Form

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

 2007-05-02