Formula:KLS:14.08:37

From DRMF
Jump to navigation Jump to search


x ⁢ Q ^ n ⁡ ( x ) = Q ^ n + 1 ⁡ ( x ) + 1 2 ⁢ ( a + b ) ⁢ q - n ⁢ Q ^ n ⁡ ( x ) + 1 4 ⁢ ( q - n - 1 ) ⁢ ( a ⁢ b ⁢ q - n + 1 - 1 ) ⁢ Q ^ n - 1 ⁡ ( x ) 𝑥 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 superscript 𝑞 1 1 2 𝑎 𝑏 superscript 𝑞 𝑛 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 1 4 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 superscript 𝑞 1 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)=% {\widehat{Q}}_{n+1}\!\left(x\right)+\frac{1}{2}(a+b)q^{-n}{\widehat{Q}}_{n}\!% \left(x\right)+\tfrac{1}{4}(q^{-n}-1)(abq^{-n+1}-1){\widehat{Q}}_{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

Q ^ n subscript ^ 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{Q}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -inverse Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqinvAlSalamChihara

Bibliography

Equation in Section 14.8 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:14.08:37&oldid=4679"