Formula:KLS:14.03:19

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formula in Continuous dual q-Hahn

Formula:KLS:14.03:21 >>

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n}){\mathcal{P}}_{n}(z)=A(z){\mathcal{P}}_{n}(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal{P}}_{n}(z)+A(z^{-1}){\mathcal{P}}_{n}(q^{-1}z)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)(1-cz)}{(1-z^{2})(1-qz^{2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle{\mathcal{P}}_{n}(z):=\frac{\left(ab,ac;q\right)_{n}}{a^{n}}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,ac}{q}{q}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.3 of KLS.

URL links

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<< Formula:KLS:14.03:15

formula in Continuous dual q-Hahn

Formula:KLS:14.03:21 >>