Formula:KLS:14.05:67

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( x - 1 ) ⁢ P n ⁡ ( x ; c ; q ) = A n ⁢ P n + 1 ⁡ ( x ; c ; q ) - ( A n + C n ) ⁢ P n ⁡ ( x ; c ; q ) + C n ⁢ P n - 1 ⁡ ( x ; c ; q ) 𝑥 1 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 subscript 𝐴 𝑛 big-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑐 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 subscript 𝐶 𝑛 big-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle(x-1)P_{n}\!\left(x;c;q\right)=A_{n}% P_{n+1}\!\left(x;c;q\right)-\left(A_{n}+C_{n}\right)P_{n}\!\left(x;c;q\right){% }+C_{n}P_{n-1}\!\left(x;c;q\right)}}}

Substitution(s)

C n = - c ⁢ q n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - c - 1 ⁢ q n ) ( 1 + q n ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝑐 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^% {-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}} &
A n = ( 1 - q n + 1 ) ⁢ ( 1 - c ⁢ q n + 1 ) ( 1 + q n + 1 ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 1 1 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{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}}}}  : big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqLegendre

Bibliography

Equation in Section 14.5 of KLS.

URL links

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