Formula:KLS:14.05:38: Difference between revisions

From DRMF
Jump to navigation Jump to search
imported>SeedBot
DRMF
 
m Move page script moved page Formula:KLS:14.05:38 to F:KLS:14.05:38
 
(No difference)

Latest revision as of 08:36, 22 December 2019


h n h 0 = q 1 2 ⁢ n ⁢ ( n - 1 ) ⁢ ( q 2 ⁢ a 2 ⁢ d c ) n ⁢ 1 - q ⁢ a ⁢ b 1 - q 2 ⁢ n + 1 ⁢ a ⁢ b ⁢ ( q , q ⁢ b , - q ⁢ b ⁢ c / d ; q ) n ( q ⁢ a , q ⁢ a ⁢ b , - q ⁢ a ⁢ d / c ; q ) n = ( - 1 ) n ⁢ ( c 2 a ⁢ b ) n ⁢ q 1 2 ⁢ n ⁢ ( n - 1 ) ⁢ q 2 ⁢ n ⁢ ( q , q ⁢ d / a , q ⁢ d / b ; q ) n ( q ⁢ c ⁢ d / ( a ⁢ b ) , q ⁢ c / a , q ⁢ c / b ; q ) n ⁢ 1 - q ⁢ c ⁢ d / ( a ⁢ b ) 1 - q 2 ⁢ n + 1 ⁢ c ⁢ d / ( a ⁢ b ) subscript ℎ 𝑛 subscript ℎ 0 superscript 𝑞 1 2 𝑛 𝑛 1 superscript superscript 𝑞 2 superscript 𝑎 2 𝑑 𝑐 𝑛 1 𝑞 𝑎 𝑏 1 superscript 𝑞 2 𝑛 1 𝑎 𝑏 q-Pochhammer-symbol 𝑞 𝑞 𝑏 𝑞 𝑏 𝑐 𝑑 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑎 𝑞 𝑎 𝑏 𝑞 𝑎 𝑑 𝑐 𝑞 𝑛 superscript 1 𝑛 superscript superscript 𝑐 2 𝑎 𝑏 𝑛 superscript 𝑞 1 2 𝑛 𝑛 1 superscript 𝑞 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑑 𝑎 𝑞 𝑑 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑐 𝑑 𝑎 𝑏 𝑞 𝑐 𝑎 𝑞 𝑐 𝑏 𝑞 𝑛 1 𝑞 𝑐 𝑑 𝑎 𝑏 1 superscript 𝑞 2 𝑛 1 𝑐 𝑑 𝑎 𝑏 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=q^{\frac{1}{2}n(% n-1)}\left(\frac{q^{2}a^{2}d}{c}\right)^{n}\frac{1-qab}{1-q^{2n+1}ab}\frac{% \left(q,qb,-qbc/d;q\right)_{n}}{\left(qa,qab,-qad/c;q\right)_{n}}=(-1)^{n}% \left(\frac{c^{2}}{ab}\right)^{n}q^{\frac{1}{2}n(n-1)}q^{2n}\frac{\left(q,qd/a% ,qd/b;q\right)_{n}}{\left(qcd/(ab),qc/a,qc/b;q\right)_{n}}\frac{1-qcd/(ab)}{1-% q^{2n+1}cd/(ab)}}}}

Substitution(s)

h 0 = ( 1 - q ) ⁢ c ⁢ ( q , - d / c , - q ⁢ c / d , q 2 ⁢ a ⁢ b ; q ) ∞ ( q ⁢ a , q ⁢ b , - q ⁢ b ⁢ c / d , - q ⁢ a ⁢ d / c ; q ) ∞ subscript ℎ 0 1 𝑞 𝑐 q-Pochhammer-symbol 𝑞 𝑑 𝑐 𝑞 𝑐 𝑑 superscript 𝑞 2 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑞 𝑎 𝑞 𝑏 𝑞 𝑏 𝑐 𝑑 𝑞 𝑎 𝑑 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/% d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}} &
h 0 = ( 1 - q ) ⁢ z + ⁢ ( q , a / c , a / d , b / c , b / d ; q ) ∞ ( a ⁢ b / ( q ⁢ c ⁢ d ) ; q ) ∞ ⁢ θ ⁢ ( z - / z + , c ⁢ d ⁢ z - ⁢ z + ; q ) θ ⁢ ( c ⁢ z - , d ⁢ z - , c ⁢ z + , d ⁢ z + ; q ) subscript ℎ 0 1 𝑞 subscript 𝑧 q-Pochhammer-symbol 𝑞 𝑎 𝑐 𝑎 𝑑 𝑏 𝑐 𝑏 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑐 𝑑 𝑞 𝜃 subscript 𝑧 subscript 𝑧 𝑐 𝑑 subscript 𝑧 subscript 𝑧 𝑞 𝜃 𝑐 subscript 𝑧 𝑑 subscript 𝑧 𝑐 subscript 𝑧 𝑑 subscript 𝑧 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a% /d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_% {-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}


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
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.5 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:14.05:38&oldid=4447"