Formula:KLS:14.07:33: Difference between revisions

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Latest revision as of 08:36, 22 December 2019


Q n ( q - x ; α , β , N ; q ) = R x ( μ ( n ) ; α , β , N ) q q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 dual-q-Hahn-R 𝑥 𝜇 𝑛 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right)=R_{x}\!\left(\mu(n);\alpha,\beta,N\right){q}}}}

Substitution(s)

μ ( x ) = q - x + γ δ q x + 1 = q - x + q x + γ + δ + 1 = q - x + γ δ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q% ^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}} &

μ ( n ) = q - n + α β q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ( x ) := q - x + γ δ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ( x ) = q - x + γ δ q x + 1 = q - x + q x + γ + δ + 1 = q - x + γ δ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q% ^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}} &

μ ( n ) = q - n + α β q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}


Proof

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Symbols List

& : logical and
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:qHahn
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : dual q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqHahn

Bibliography

Equation in Section 14.7 of KLS.

URL links

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