Formula:KLS:09.08:13: Difference between revisions

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Latest revision as of 08:35, 22 December 2019


R - 1 ⁢ ( 1 + R - x ⁢ t 2 ) 1 2 - λ = ∑ n = 0 ∞ ( λ + 1 2 ) n ( 2 ⁢ λ ) n ⁢ C n λ ⁡ ( x ) ⁢ t n superscript 𝑅 1 superscript 1 𝑅 𝑥 𝑡 2 1 2 𝜆 superscript subscript 𝑛 0 Pochhammer-symbol 𝜆 1 2 𝑛 Pochhammer-symbol 2 𝜆 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)% ^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty}\frac{{\left(\lambda+\frac{1}{2}% \right)_{n}}}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}}

Substitution(s)

R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


Proof

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Symbols List

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

Bibliography

Equation in Section 9.8 of KLS.

URL links

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