Formula:KLS:14.05:64

<< Formula:KLS:14.05:63

formula in Big q-Jacobi

Formula:KLS:14.05:65 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}P_{n}\!\left(\mathrm{i}q^{\frac{1}{2}(-N-1+\mathrm{i}\nu)}x;-q^{-N-1},-q^{-N-1},q^{-N+\mathrm{i}\nu-1};q\right)=\frac{P_{n}\!\left(x;\nu,N\right)}{P_{n}\!\left(-\mathrm{i};\nu,N\right)}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : pseudo Jacobi polynomomal : http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:63

formula in Big q-Jacobi

Formula:KLS:14.05:65 >>