Formula:KLS:14.08:32

$\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamChihara{n}@{\cos@{\ln@@{q^x}+\phi}}{ q^{\lambda}\expe^{\iunit\phi}}{q^{\lambda}\expe^{-\iunit\phi}}{q}}{\qPochhammer{q}{q}{n}}=\MeixnerPollaczek{\lambda}{n}@{x}{\phi} }$

Proof

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Symbols List

$\displaystyle {\displaystyle Q_{n}}$  : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2
$\displaystyle {\displaystyle \mathrm{ln}}$  : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
$\displaystyle {\displaystyle \mathrm{e}}$  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle P^{(\alpha)}_{n}}$  : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1