Confluent hypergeometric function 0F1: Representations through more general functions (formula 07.17.26.0139)

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/07.17.26.0139.01

Input Form

Hypergeometric0F1[b, -(z^2/4)] BesselY[-b - n, 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)}}, z^2] - ((2^b z^(-b - n) Cot[b Pi] Gamma[b])/ Sqrt[Pi]) Sum[(((-1)^(k + Floor[(1 + n)/2]) z^(2 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 && Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]

Standard Form

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MathML Form

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type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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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]