Formula:KLS:14.12:41

<< Formula:KLS:14.12:40

formula in Little q-Jacobi: Special case

Formula:KLS:14.12:42 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{0}{1}@@{-}{q}{q}{qxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{q}{q}{xt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q,q;q\right)_{n}}p_{n}\!\left(x|q\right)t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqLegendre

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:40

formula in Little q-Jacobi: Special case

Formula:KLS:14.12:42 >>