html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 AppellF1

 http://functions.wolfram.com/07.36.06.0009.01

 Input Form

 AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] == ((Gamma[c] Gamma[c - a - Subscript[b, 1]])/ Gamma[c - a]) Sum[((Pochhammer[a, k] Pochhammer[Subscript[b, 2], k])/ (k! Gamma[c + k - Subscript[b, 1]])) Hypergeometric2F1[a + k, Subscript[b, 1], 1 + a - c + Subscript[b, 1], 1 - Subscript[z, 1]] Subscript[z, 2]^k, {k, 0, Infinity}] + ((Gamma[c] Gamma[a - c + Subscript[b, 1]])/ (Gamma[a] Gamma[Subscript[b, 1]])) (1 - Subscript[z, 1])^ (c - a - Subscript[b, 1]) Sum[(Pochhammer[Subscript[b, 2], k]/k!) Hypergeometric2F1[c - a, c - Subscript[b, 1] + k, 1 - a + c - Subscript[b, 1], 1 - Subscript[z, 1]] Subscript[z, 2]^k, {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", RowBox[List["c", "-", "a", "-", SubscriptBox["b", "1"]]], "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]]]], RowBox[List[" ", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "+", "k", "-", SubscriptBox["b", "1"]]], "]"]]]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["a", "+", "k"]], ",", SubscriptBox["b", "1"], ",", RowBox[List["1", "+", "a", "-", "c", "+", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["z", "1"]]]]], "]"]], " ", SubsuperscriptBox["z", "2", "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "c", "+", SubscriptBox["b", "1"]]], "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["c", "-", "a", "-", SubscriptBox["b", "1"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]], RowBox[List[" ", RowBox[List["k", "!"]], " "]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", RowBox[List["c", "-", SubscriptBox["b", "1"], "+", "k"]], ",", RowBox[List["1", "-", "a", "+", "c", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["z", "1"]]]]], "]"]], " ", SubsuperscriptBox["z", "2", "k"]]]]]]]]]]]]]

 MathML Form

 F 1 AppellF1 ( a ; b 1 , b 2 ; c ; z 1 , z 2 ) Γ ( c ) Γ ( a - c + b 1 ) Γ ( a ) Γ ( b 1 ) ( 1 - z 1 ) c - a - b 1 k = 0 ( b 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "k"], Pochhammer] k ! 2 F 1 ( c - a , c + k - b 1 ; c - a - b 1 + 1 ; 1 - z 1 ) 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[RowBox[List["c", "-", "a"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "+", "k", "-", SubscriptBox["b", "1"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "a", "-", SubscriptBox["b", "1"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", SubscriptBox["z", "1"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] z 2 k + Γ ( c ) Γ ( c - a - b 1 ) Γ ( c - a ) k = 0 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( b 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "k"], Pochhammer] k ! Γ ( c + k - b 1 ) 2 F 1 ( a + k , b 1 ; a - c + b 1 + 1 ; 1 - z 1 ) 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[RowBox[List["a", "+", "k"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "1"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "-", "c", "+", SubscriptBox["b", "1"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", SubscriptBox["z", "1"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] z 2 k AppellF1 a Subscript b 1 Subscript b 2 c Subscript z 1 Subscript z 2 Gamma c Gamma a -1 c Subscript b 1 Gamma a Gamma Subscript b 1 -1 1 -1 Subscript z 1 c -1 a -1 Subscript b 1 k 0 Pochhammer Subscript b 2 k k -1 Hypergeometric2F1 c -1 a c k -1 Subscript b 1 c -1 a -1 Subscript b 1 1 1 -1 Subscript z 1 Subscript z 2 k Gamma c Gamma c -1 a -1 Subscript b 1 Gamma c -1 a -1 k 0 Pochhammer a k Pochhammer Subscript b 2 k k Gamma c k -1 Subscript b 1 -1 Hypergeometric2F1 a k Subscript b 1 a -1 c Subscript b 1 1 1 -1 Subscript z 1 Subscript z 2 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AppellF1", "[", RowBox[List["a_", ",", SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", "c_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "a", "-", SubscriptBox["bb", "1"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["a", "+", "k"]], ",", SubscriptBox["bb", "1"], ",", RowBox[List["1", "+", "a", "-", "c", "+", SubscriptBox["bb", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["zz", "1"]]]]], "]"]], " ", SubsuperscriptBox["zz", "2", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "+", "k", "-", SubscriptBox["bb", "1"]]], "]"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "c", "+", SubscriptBox["bb", "1"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["c", "-", "a", "-", SubscriptBox["bb", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", RowBox[List["c", "-", SubscriptBox["bb", "1"], "+", "k"]], ",", RowBox[List["1", "-", "a", "+", "c", "-", SubscriptBox["bb", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["zz", "1"]]]]], "]"]], " ", SubsuperscriptBox["zz", "2", "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]]]]]]]]]]]

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

 2001-10-29