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; }

 MeijerG

 http://functions.wolfram.com/07.34.13.0004.01

 Input Form

 (-1)^(m + n - p) z Fold[Function[{f, l}, z D[f, z] + (1 - Subscript[a, l]) f], w[z], {1, \[Ellipsis], p}] - Fold[Function[{f, k}, z D[f, z] - Subscript[b, k] f], w[z], {1, \[Ellipsis], q}] == 0 /; w[z] == Sum[Subscript[c, k] z^Subscript[b, k] HypergeometricPFQ[ {1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[a, p]}, {1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] - Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, q]}, (-1)^(p - m - n) z], {k, 1, q}] && (p < q || (p == q && Abs[z] < 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= q && 1 <= k <= q, !Element[Subscript[b, j] - Subscript[b, k], Integers]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n", "-", "p"]]], " ", "z", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "l"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "l"]]], ")"]], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "p"]], "}"]]]], "]"]]]], "-", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "-", RowBox[List[SubscriptBox["b", "k"], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "q"]], "}"]]]], "]"]]]], "\[Equal]", "0"]], " ", "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SubscriptBox["c", "k"], SuperscriptBox["z", SubscriptBox["b", "k"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]]]], "]"]]]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["p", "<", "q"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], ")"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "q"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "q"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]

 MathML Form

 ( ( - 1 ) m + n - p z l = 1 p ( z d d z + 1 - a l ) - k = 1 q ( z d d z - b k ) ) w ( z ) 0 /; w ( z ) k = 1 q c k z b k p F q - 1 ( 1 - a 1 + b k , , 1 - a p + b k ; 1 - b 1 + b k , , 1 - b k - 1 + b k , 1 - b k + 1 + b k , , 1 - b q + b k ; ( - 1 ) p - m - n z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], SubscriptBox["F", RowBox[List["q", "-", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( p < q ( p q "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 ) ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] j k 1 j q 1 k q ( b j - b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) Condition -1 m n -1 p z l 1 p z d d z -1 1 -1 Subscript a l -1 k 1 q z d d z -1 -1 Subscript b k w z 0 w z k 1 q Subscript c k z Subscript b k HypergeometricPFQ 1 -1 Subscript a 1 Subscript b k 1 -1 Subscript a p Subscript b k 1 -1 Subscript b 1 Subscript b k 1 -1 Subscript b k -1 Subscript b k 1 -1 Subscript b k 1 Subscript b k 1 -1 Subscript b q Subscript b k -1 p -1 m -1 n z p q p q z 1 j k j k j k 1 j q 1 k q Subscript b j -1 Subscript b k [/itex]

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

 2007-05-02