Formula:KLS:14.04:26

<< Formula:KLS:14.04:25

formula in Continuous q-Hahn

Formula:KLS:14.04:27 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{p_{n}\!\left(\cos\left(\ln q^{-x}+\phi\right);q^{a},q^{b},q^{c},q^{d};q\right)}{(1-q)^{n}\left(q;q\right)_{n}}=(-2\sin\phi)^{n}p_{n}\!\left(x;a,b,c,d\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1

Bibliography

Equation in Section 14.4 of KLS.

URL links

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formula in Continuous q-Hahn

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