Formula:KLS:14.04:07

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2 x p ~ n ( x ) = A n p ~ n + 1 ( x ) + [ a e i ϕ + a - 1 e - i ϕ - ( A n + C n ) ] p ~ n ( x ) + C n p ~ n - 1 ( x ) 2 𝑥 continuous-q-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐴 𝑛 continuous-q-Hahn-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 delimited-[] 𝑎 imaginary-unit italic-ϕ superscript 𝑎 1 imaginary-unit italic-ϕ subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-q-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐶 𝑛 continuous-q-Hahn-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle 2x{\tilde{p}}_{n}\!\left(x\right)=A% _{n}{\tilde{p}}_{n+1}\!\left(x\right)+\left[a{\mathrm{e}^{\mathrm{i}\phi}}+a^{% -1}{\mathrm{e}^{-\mathrm{i}\phi}}-\left(A_{n}+C_{n}\right)\right]{\tilde{p}}_{% n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = a e i ϕ ( 1 - q n ) ( 1 - b c q n - 1 ) ( 1 - b d q n - 1 ) ( 1 - c d e - 2 i ϕ q n - 1 ) ( 1 - a b c d q 2 n - 2 ) ( 1 - a b c d q 2 n - 1 ) subscript 𝐶 𝑛 𝑎 imaginary-unit italic-ϕ 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 1 𝑏 𝑑 superscript 𝑞 𝑛 1 1 𝑐 𝑑 2 imaginary-unit italic-ϕ superscript 𝑞 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 2 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a{\mathrm{e}^{\mathrm{i% }\phi}}(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cd{\mathrm{e}^{-2\mathrm{i}\phi}}% q^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}} &
A n = ( 1 - a b e 2 i ϕ q n ) ( 1 - a c q n ) ( 1 - a d q n ) ( 1 - a b c d q n - 1 ) a e i ϕ ( 1 - a b c d q 2 n - 1 ) ( 1 - a b c d q 2 n ) subscript 𝐴 𝑛 1 𝑎 𝑏 2 imaginary-unit italic-ϕ superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 1 𝑎 𝑑 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 imaginary-unit italic-ϕ 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-ab{\mathrm{e}^{2% \mathrm{i}\phi}}q^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a{\mathrm{e}^{% \mathrm{i}\phi}}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}


Proof

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Symbols List

& : logical and
p ~ n subscript ~ 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}}}}  : normalized continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial p ~ ~ 𝑝 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}}}}  : http://drmf.wmflabs.org/wiki/Definition:normctsqHahnptilde
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

Bibliography

Equation in Section 14.4 of KLS.

URL links

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