Formula:KLS:14.06:16

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

Substitution(s)

w ⁢ ( x ; α , β , N | q ) = ( α ⁢ q , q - N ; q ) x ( q , β - 1 ⁢ q - N ; q ) x ⁢ ( α ⁢ β ) - x 𝑤 𝑥 𝛼 𝛽 conditional 𝑁 𝑞 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛽 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝛼 𝛽 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\left(% \alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha% \beta)^{-x}}}}


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
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.6 of KLS.

URL links

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