Formula:KLS:01.14:07

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Cos q ( z ) := E q ( i z ) + E q ( - i z ) 2 = n = 0 ( - 1 ) n q \binomial 2 n 2 z 2 n ( q ; q ) 2 n assign KLS-q-Cos 𝑞 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 2 𝑛 2 superscript 𝑧 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}\!\left(z\right):=% \frac{\mathrm{E}_{q}\!\left(\mathrm{i}z\right)+\mathrm{E}_{q}\!\left(-\mathrm{% i}z\right)}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{2n}{2}}z^{2n}}{% \left(q;q\right)_{2n}}}}}

Proof

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Symbols List

Cos q subscript Cos 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -analogue of the cosine function Cos q subscript Cos 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}}}} used in KLS : http://drmf.wmflabs.org/wiki/Definition:qCosKLS
E q subscript E 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -analogue of the exponential function E q subscript E 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}} used in KLS : http://drmf.wmflabs.org/wiki/Definition:qExpKLS
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( n k ) binomial 𝑛 𝑘 {\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
( 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

Bibliography

Equation in Section 1.14 of KLS.

URL links

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