Formula:KLS:14.10:72: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


lim β 1 1 - q n 2 ( 1 - β ) C n ( x ; β | q ) = cos n θ = T n ( x ) , x = cos θ formulae-sequence subscript 𝛽 1 1 superscript 𝑞 𝑛 2 1 𝛽 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 𝑛 𝜃 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 1}\frac{1-q^{% n}}{2(1-\beta)}C_{n}\!\left(x;\beta\,|\,q\right)=\cos n\theta=T_{n}\left(x% \right),\quad x=\cos\theta}}}

Constraint(s)

n = 1 , 2 , 3 , 𝑛 1 2 3 {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}


Proof

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Symbols List

C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2
T n subscript 𝑇 𝑛 {\displaystyle{\displaystyle{\displaystyle T_{n}}}}  : Chebyshev polynomial of the first kind : http://dlmf.nist.gov/18.3#T1.t1.r8

Bibliography

Equation in Section 14.10 of KLS.

URL links

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