Formula:KLS:14.06:03: Difference between revisions

From DRMF
Jump to navigation Jump to search
imported>SeedBot
DRMF
 
m Move page script moved page Formula:KLS:14.06:03 to F:KLS:14.06:03
 
(No difference)

Latest revision as of 08:36, 22 December 2019


- ( 1 - q - x ) ⁢ Q n ⁡ ( q - x ) = A n ⁢ Q n + 1 ⁡ ( q - x ) - ( A n + C n ) ⁢ Q n ⁡ ( q - x ) + C n ⁢ Q n - 1 ⁡ ( q - x ) 1 superscript 𝑞 𝑥 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐶 𝑛 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)Q_{n}\!\left(q% ^{-x}\right)=A_{n}Q_{n+1}\!\left(q^{-x}\right)-\left(A_{n}+C_{n}\right)Q_{n}\!% \left(q^{-x}\right){}+C_{n}Q_{n-1}\!\left(q^{-x}\right)}}}

Substitution(s)

C n = - α ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + N + 1 ) ⁢ ( 1 - β ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝛼 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 𝑁 1 1 𝛽 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-\frac{\alpha q^{n-N}(1-q^{n}% )(1-\alpha\beta q^{n+N+1})(1-\beta q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha% \beta q^{2n+1})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1-\alpha q^% {n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2}% )}}}}


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

& : logical and
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

Bibliography

Equation in Section 14.6 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:14.06:03&oldid=4523"