Formula:KLS:14.12:31: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


- x ⁢ p n ⁡ ( x | q ) = A n ⁢ p n + 1 ⁡ ( x | q ) - ( A n + C n ) ⁢ p n ⁡ ( x | q ) + C n ⁢ p n - 1 ⁡ ( x | q ) 𝑥 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝐴 𝑛 little-q-Legendre-polynomial-p 𝑛 1 𝑥 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝐶 𝑛 little-q-Legendre-polynomial-p 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x|q\right)=A_{n}p_{n+% 1}\!\left(x|q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x|q\right)+C_{n}p_{n% -1}\!\left(x|q\right)}}}

Substitution(s)

C n = q n ⁢ ( 1 - q n ) ( 1 + q n ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=q^{n}\frac{(1-q^{n})}{(1+q^{n% })(1-q^{2n+1})}}}} &
A n = q n ⁢ ( 1 - q n + 1 ) ( 1 + q n + 1 ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1-q^{n+1})}{(1+q^% {n+1})(1-q^{2n+1})}}}}


Proof

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Symbols List

& : logical and
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqLegendre

Bibliography

Equation in Section 14.12 of KLS.

URL links

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