Formula:KLS:14.13:07

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- ( 1 - q n ) y ( x ) = B ( x ) y ( x + 1 ) - [ B ( x ) + D ( x ) ] y ( x ) + D ( x ) y ( x - 1 ) 1 superscript 𝑞 𝑛 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 1 {\displaystyle{\displaystyle{\displaystyle-(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)% +D(x)\right]y(x)+D(x)y(x-1)}}}

Substitution(s)

D ( x ) = ( 1 - q x ) ( 1 + b c q x ) 𝐷 𝑥 1 superscript 𝑞 𝑥 1 𝑏 𝑐 superscript 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle D(x)=(1-q^{x})(1+bcq^{x})}}} &

B ( x ) = c q x ( 1 - b q x + 1 ) 𝐵 𝑥 𝑐 superscript 𝑞 𝑥 1 𝑏 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=cq^{x}(1-bq^{x+1})}}} &

y ( x ) = M n ( q - x ; b , c ; q ) 𝑦 𝑥 q-Meixner-polynomial-M 𝑛 superscript 𝑞 𝑥 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=M_{n}\!\left(q^{-x};b,c;q% \right)}}}


Proof

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Symbols List

& : logical and
M n subscript 𝑀 𝑛 {\displaystyle{\displaystyle{\displaystyle M_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Meixner polynomial : http://drmf.wmflabs.org/wiki/Definition:qMeixner

Bibliography

Equation in Section 14.13 of KLS.

URL links

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