Formula:KLS:09.08:45

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T n ⁡ ( x ) = P n ( - 1 2 , - 1 2 ) ⁡ ( x ) P n ( - 1 2 , - 1 2 ) ⁡ ( 1 ) = \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n ⁢ 1 2 ⁢ 1 - x 2 formulae-sequence Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 \HyperpFq 21 @ @ 𝑛 𝑛 1 2 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle T_{n}\left(x\right)=\frac{P^{(-% \frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_% {n}\left(1\right)}=\HyperpFq{2}{1}@@{-n,n}{\frac{1}{2}}{\frac{1-x}{2}}}}}

Proof

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Symbols List

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
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3
F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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