Formula:KLS:14.08:31

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Q n ( x ; q 1 2 α + 1 4 , q 1 2 α + 3 4 | q ) = ( q ; q ) n q ( 1 2 α + 1 4 ) n P n ( α ) ( x | q ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 3 4 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;q^{\frac{1}{2}\alpha% +\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}}\,|\,q\right)=\frac{\left(q;q% \right)_{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha)}_{n}\!\left(x|q% \right)}}}

Proof

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Symbols List

Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
P α ( n ) subscript superscript 𝑃 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle P^{(n)}_{\alpha}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre

Bibliography

Equation in Section 14.8 of KLS.

URL links

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