Formula:KLS:14.23:06

From DRMF
Jump to navigation Jump to search


x ⁢ C ^ n ⁡ ( x ) = C ^ n + 1 ⁡ ( x ) + [ 1 + q - 2 ⁢ n - 1 ⁢ { a + q ⁢ ( 1 - q n ) ⁢ ( a + q n ) } ] ⁢ C ^ n ⁡ ( x ) + a ⁢ q - 4 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ ( a + q n ) ⁢ C ^ n - 1 ⁡ ( x ) 𝑥 q-Charlier-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 q-Charlier-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 delimited-[] 1 superscript 𝑞 2 𝑛 1 𝑎 𝑞 1 superscript 𝑞 𝑛 𝑎 superscript 𝑞 𝑛 q-Charlier-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 4 𝑛 1 1 superscript 𝑞 𝑛 𝑎 superscript 𝑞 𝑛 q-Charlier-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}_{n}\!\left(x\right)=% {\widehat{C}}_{n+1}\!\left(x\right)+\left[1+q^{-2n-1}\left\{a+q(1-q^{n})(a+q^{% n})\right\}\right]{\widehat{C}}_{n}\!\left(x\right){}+aq^{-4n+1}(1-q^{n})(a+q^% {n}){\widehat{C}}_{n-1}\!\left(x\right)}}}

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

C ^ n subscript ^ 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{C}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Charlier polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqCharlier

Bibliography

Equation in Section 14.23 of KLS.

URL links

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:14.23:06&oldid=5479"