Confluent hypergeometric function 0F1: Representations through more general functions (formula 07.17.26.0131)

Hypergeometric0F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric0F1[b,z]

Representations through more general functions

Through Meijer G

Classical cases involving Bessel Y

http://functions.wolfram.com/07.17.26.0131.01

Input Form

Hypergeometric0F1[b, -z] BesselY[-b - n, 2 Sqrt[z]] == (((-1)^n 2^(-1 + b) Gamma[b])/Sqrt[Pi]) MeijerG[{{(1 - b)/2, 1 - b/2}, {(1/2) (1 + b + n)}}, {{(1/2) (2 - b + n), (b + n)/2}, {(1/2) (2 - 3 b - n), (1/2) (-b - n), (1/2) (1 + b + n)}}, 4 z] - ((z^((1/2) (-b - n)) Cot[b Pi] Gamma[b])/ (2^n Sqrt[Pi])) Sum[(((-1)^(k + Floor[(1 + n)/2]) 4^k z^k Gamma[1/2 + k - n + Floor[n/2]])/(k! Gamma[b + k] Gamma[1 - b + k - n])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0

Standard Form

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MathML Form

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<ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", RowBox[List["-", "z_"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "b_"]], "-", "n_"]], ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "b"]]], " ", RowBox[List["Gamma", "[", "b", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "b", "+", "n"]], ")"]]]], ",", FractionBox[RowBox[List["b", "+", "n"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "b"]], "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], SqrtBox["\[Pi]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Cot", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["z", "k"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "+", "k", "-", "n"]], "]"]]]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]