Chebyshev function of the second kind: Series representations (formula 07.05.06.0077)

ChebyshevU

$Mathematica Notation$

Hypergeometric Functions

ChebyshevU[nu,z]

Series representations

Generalized power series

Expansions at z==infinity

For the function itself

Expansions in 1/z

http://functions.wolfram.com/07.05.06.0077.01

Input Form

ChebyshevU[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; (Subscript[F, m][z, \[Nu]] == 2^\[Nu] z^\[Nu] Sum[(Pochhammer[-(\[Nu]/2), k] Pochhammer[(1 - \[Nu])/2, k])/(Pochhammer[-\[Nu], k] k!)/z^(2 k), {k, 0, m}] - 2^(-2 - \[Nu]) z^(-2 - \[Nu]) Sum[(Pochhammer[(3 + \[Nu])/2, k] Pochhammer[(2 + \[Nu])/2, k])/ (Pochhammer[2 + \[Nu], k] k!)/z^(2 k), {k, 0, m}] == ChebyshevU[\[Nu], z] - ((2^(-2 - 2 m + \[Nu]) z^(-2 - 2 m + \[Nu]) Gamma[2 + 2 m - \[Nu]])/(Gamma[2 + m] Gamma[1 + m - \[Nu]])) HypergeometricPFQ[{1, 1 + m - \[Nu]/2, 3/2 + m - \[Nu]/2}, {2 + m, 1 + m - \[Nu]}, 1/z^2] + ((2^(-4 - 2 m - \[Nu]) z^(-4 - 2 m - \[Nu]) Gamma[4 + 2 m + \[Nu]])/ (Gamma[2 + m] Gamma[3 + m + \[Nu]])) HypergeometricPFQ[ {1, 2 + m + \[Nu]/2, 5/2 + m + \[Nu]/2}, {2 + m, 3 + m + \[Nu]}, 1/z^2] && Element[m, Integers] && m >= 0) && !Element[\[Nu], Integers]

Standard Form

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MathML Form

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Condition </ci> <apply> <eq /> <apply> <ci> ChebyshevU </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> F </ci> <infinity /> </apply> <ci> z </ci> <ci> ν </ci> </apply> </apply> <apply> <and /> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> F </ci> <ci> m </ci> </apply> <ci> z </ci> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> ChebyshevU </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -4 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -4 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </list> <list> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> m </ci> </apply> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> m </ci> </apply> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <list> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> </apply> <apply> <notin /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

ChebyshevU[n,z]