Formula:KLS:14.06:16

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formula in q-Hahn

Formula:KLS:14.06:18 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;\alpha,\beta,N|q)\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}\right]}{\nabla q^{-x}}{}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N}}{q}{x}}(\alpha\beta)^{-x}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:qHahn
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.6 of KLS.

URL links

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<< Formula:KLS:14.06:15

formula in q-Hahn

Formula:KLS:14.06:18 >>