Formula:KLS:14.04:16

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formula in Continuous q-Hahn

Formula:KLS:14.04:17 >>

${\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c,d;q\right){}=-q^{-\frac{1}{2}n}(1-q^{n})(1-abcdq^{n-1})({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}};q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.4 of KLS.

URL links

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