Formula:KLS:14.06:15

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( 1 - α ⁢ q x ) ⁢ ( 1 - q x - N - 1 ) ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) - α ⁢ ( 1 - q x ) ⁢ ( β - q x - N - 1 ) ⁢ Q n ⁡ ( q - x + 1 ; α , β , N ; q ) = q x ⁢ ( 1 - α ) ⁢ ( 1 - q - N - 1 ) ⁢ Q n + 1 ⁢ ( q - x ; α ⁢ q - 1 , β ⁢ q - 1 , N + 1 | q ) 1 𝛼 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 𝑁 1 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 𝛼 1 superscript 𝑞 𝑥 𝛽 superscript 𝑞 𝑥 𝑁 1 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 1 𝛼 𝛽 𝑁 𝑞 superscript 𝑞 𝑥 1 𝛼 1 superscript 𝑞 𝑁 1 subscript 𝑄 𝑛 1 superscript 𝑞 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝑁 conditional 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-\alpha q^{x})(1-q^{x-N-1})Q_{n}\!% \left(q^{-x};\alpha,\beta,N;q\right){}-\alpha(1-q^{x})(\beta-q^{x-N-1})Q_{n}\!% \left(q^{-x+1};\alpha,\beta,N;q\right){}=q^{x}(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^% {-x};\alpha q^{-1},\beta q^{-1},N+1|q)}}}

Proof

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Symbols List

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

Bibliography

Equation in Section 14.6 of KLS.

URL links

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