Formula:KLS:14.13:05: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


x ⁢ M ^ n ⁡ ( x ) = M ^ n + 1 ⁡ ( x ) + [ 1 + q - 2 ⁢ n - 1 ⁢ { c ⁢ ( 1 - b ⁢ q n + 1 ) + q ⁢ ( 1 - q n ) ⁢ ( c + q n ) } ] ⁢ M ^ n ⁡ ( x ) + c ⁢ q - 4 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ q n ) ⁢ ( c + q n ) ⁢ M ^ n - 1 ⁡ ( x ) 𝑥 q-Meixner-polynomial-monic-M 𝑛 𝑥 𝑏 𝑐 𝑞 q-Meixner-polynomial-monic-M 𝑛 1 𝑥 𝑏 𝑐 𝑞 delimited-[] 1 superscript 𝑞 2 𝑛 1 𝑐 1 𝑏 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑛 𝑐 superscript 𝑞 𝑛 q-Meixner-polynomial-monic-M 𝑛 𝑥 𝑏 𝑐 𝑞 𝑐 superscript 𝑞 4 𝑛 1 1 superscript 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 𝑐 superscript 𝑞 𝑛 q-Meixner-polynomial-monic-M 𝑛 1 𝑥 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{M}}_{n}\!\left(x\right)=% {\widehat{M}}_{n+1}\!\left(x\right)+\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-% q^{n})(c+q^{n})\right\}\right]{\widehat{M}}_{n}\!\left(x\right){}+cq^{-4n+1}(1% -q^{n})(1-bq^{n})(c+q^{n}){\widehat{M}}_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

M ^ n subscript ^ 𝑀 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{M}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Meixner polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqMeixner

Bibliography

Equation in Section 14.13 of KLS.

URL links

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