Formula:KLS:14.10:08

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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q}=A_n\normctsqJacobiPtilde{n+1}{\beta}{n}@@{x}{q}+\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_n+C_n\right)\right]\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q} {}+C_n\normctsqJacobiPtilde{n-1}{\beta}{n}@@{x}{q} }}

Substitution(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^n)(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})} {(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})} {q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}


Proof

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Symbols List

& : logical and
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde P}^{(\alpha,\beta)}_{n}}}  : normalized continuous Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -Jacobi polynomial Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde P}}}  : http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde

Bibliography

Equation in Section 14.10 of KLS.

URL links

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