Jacobi function: Series representations (formula 07.15.06.0030)

JacobiP

$Mathematica Notation$

Hypergeometric Functions

JacobiP[nu,a,b,z]

Series representations

Generalized power series

Expansions at z==infinity

http://functions.wolfram.com/07.15.06.0030.01

Input Form

JacobiP[\[Nu], a, b, z] == (((-1)^(a + \[Nu]) 2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] (b + \[Nu])! (a + \[Nu])!)/(Pi (1 + a + b + 2 \[Nu])!)) (z - 1)^(-1 - a - b - \[Nu]) Hypergeometric2F1[1 + a + b + \[Nu], 1 + b + \[Nu], 2 + a + b + 2 \[Nu], 2/(1 - z)] + ((a + b + 2 \[Nu])!/(2^\[Nu] (Gamma[1 + \[Nu]] Gamma[1 + a + b + \[Nu]]))) (z - 1)^\[Nu] Sum[((Pochhammer[-a - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! Pochhammer[-a - b - 2 \[Nu], k])) (2/(1 - z))^k, {k, 0, a + \[Nu]}] /; Element[1 + a + b + 2 \[Nu], Integers] && 1 + a + b + 2 \[Nu] >= 0 && Element[a + \[Nu], Integers] && a + \[Nu] >= 0 && b + \[Nu] >= 0 && !IntervalMemberQ[Interval[{-1, 1}], z]

Standard Form

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MathML Form

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</ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ν </ms> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> b </ms> <ms> + </ms> <ms> ν </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> ν </ms> </list> </apply> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> a </ms> </list> </apply> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> a </ms> </list> </apply> <ms> - </ms> <ms> b </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> ν </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> b </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> ν </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ∈ </ms> <ms> ℕ </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> ν </ms> </list> </apply> <ms> ∈ </ms> <ms> ℕ </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <ms> + </ms> <ms> ν </ms> </list> </apply> <ms> ≥ </ms> <ms> 0 </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ∉ </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <ms> 1 </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

JacobiP[n,a,b,z]