Formula:KLS:14.03:21

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δ q p n ( x ; a , b , c | q ) = - q - 1 2 n ( 1 - q n ) ( e i θ - e - i θ ) p n - 1 ( x ; a q 1 2 , b q 1 2 , c q 1 2 | q ) subscript 𝛿 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 imaginary-unit 𝜃 imaginary-unit 𝜃 continuous-dual-q-Hahn-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c|q% \right)=-q^{-\frac{1}{2}n}(1-q^{n})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e% }^{-\mathrm{i}\theta}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq% ^{\frac{1}{2}}|q\right)}}}

Substitution(s)

x = cos θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Proof

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Symbols List

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

Bibliography

Equation in Section 14.3 of KLS.

URL links

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