Formula:KLS:14.02:05: Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
imported>SeedBot
DRMF
m Move page script moved page Formula:KLS:14.02:05 to F:KLS:14.02:05
 
(No difference)

Latest revision as of 08:36, 22 December 2019


∑ x = 0 N ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q , γ ⁢ δ ⁢ q ; q ) x ( q , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) ( α ⁢ β ⁢ q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q ) ⁢ R m ⁡ ( μ ⁢ ( x ) ) ⁢ R n ⁡ ( μ ⁢ ( x ) ) = h n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 superscript 𝛼 𝛽 𝑞 𝑥 1 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑚 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\alpha q,% \beta\delta q,\gamma q,\gamma\delta q;q\right)_{x}}{\left(q,\alpha^{-1}\gamma% \delta q,\beta^{-1}\gamma q,\delta q;q\right)_{x}}{}\frac{(1-\gamma\delta q^{2% x+1})}{(\alpha\beta q)^{x}(1-\gamma\delta q)}R_{m}\!\left(\mu(x)\right)R_{n}\!% \left(\mu(x)\right)=h_{n}\,\delta_{m,n}}}}

Substitution(s)

μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=% \lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
h n = ( α - 1 ⁢ β - 1 ⁢ γ , α - 1 ⁢ δ , β - 1 , γ ⁢ δ ⁢ q 2 ; q ) ∞ ( α - 1 ⁢ β - 1 ⁢ q - 1 , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) ∞ ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( γ ⁢ δ ⁢ q ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ γ - 1 ⁢ q , α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q ) n = { ( β - 1 , γ ⁢ δ ⁢ q 2 ; q ) N ( β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) N ⁢ ( 1 - β ⁢ q - N ) ⁢ ( γ ⁢ δ ⁢ q ) n ( 1 - β ⁢ q 2 ⁢ n - N ) ⁢ ( q , β ⁢ q , β ⁢ γ - 1 ⁢ q - N , δ - 1 ⁢ q - N ; q ) n ( β ⁢ q - N , β ⁢ δ ⁢ q , γ ⁢ q , q - N ; q ) n < b r / >   if   α q = q - N ( α ⁢ β ⁢ q 2 , β ⁢ γ - 1 ; q ) N ( α ⁢ β ⁢ γ - 1 ⁢ q , β ⁢ q ; q ) N ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( β - 1 ⁢ γ ⁢ q - N ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ q N + 2 , α ⁢ β ⁢ γ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , γ ⁢ q , q - N ; q ) n < b r / >   if   β δ q = q - N ( α ⁢ β ⁢ q 2 , δ - 1 ; q ) N ( α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) N ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( δ ⁢ q - N ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ q N + 2 , α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , β ⁢ δ ⁢ q , q - N ; q ) n < b r / >   if   γ q = q - N subscript ℎ 𝑛 q-Pochhammer-symbol superscript 𝛼 1 superscript 𝛽 1 𝛾 superscript 𝛼 1 𝛿 superscript 𝛽 1 𝛾 𝛿 superscript 𝑞 2 𝑞 q-Pochhammer-symbol superscript 𝛼 1 superscript 𝛽 1 superscript 𝑞 1 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 1 𝛼 𝛽 𝑞 superscript 𝛾 𝛿 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 cases q-Pochhammer-symbol superscript 𝛽 1 𝛾 𝛿 superscript 𝑞 2 𝑞 𝑁 q-Pochhammer-symbol superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑁 1 𝛽 superscript 𝑞 𝑁 superscript 𝛾 𝛿 𝑞 𝑛 1 𝛽 superscript 𝑞 2 𝑛 𝑁 q-Pochhammer-symbol 𝑞 𝛽 𝑞 𝛽 superscript 𝛾 1 superscript 𝑞 𝑁 superscript 𝛿 1 superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝛽 superscript 𝑞 𝑁 𝛽 𝛿 𝑞 𝛾 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  α q superscript 𝑞 𝑁 missing-subexpression missing-subexpression q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 2 𝛽 superscript 𝛾 1 𝑞 𝑁 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛽 𝑞 𝑞 𝑁 1 𝛼 𝛽 𝑞 superscript superscript 𝛽 1 𝛾 superscript 𝑞 𝑁 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝑞 𝑁 2 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛾 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  β δ q superscript 𝑞 𝑁 missing-subexpression missing-subexpression q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 2 superscript 𝛿 1 𝑞 𝑁 q-Pochhammer-symbol 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑁 1 𝛼 𝛽 𝑞 superscript 𝛿 superscript 𝑞 𝑁 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝑞 𝑁 2 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  γ q superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(\alpha^{-1}\beta^% {-1}\gamma,\alpha^{-1}\delta,\beta^{-1},\gamma\delta q^{2};q\right)_{\infty}}{% \left(\alpha^{-1}\beta^{-1}q^{-1},\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q% ,\delta q;q\right)_{\infty}}{}\frac{(1-\alpha\beta q)(\gamma\delta q)^{n}}{(1-% \alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta\gamma^{-1}q,\alpha\delta^{-1}q% ,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,\gamma q;q% \right)_{n}}=\left\{\begin{array}[]{ll}\displaystyle\frac{\left(\beta^{-1},% \gamma\delta q^{2};q\right)_{N}}{\left(\beta^{-1}\gamma q,\delta q;q\right)_{N% }}\frac{(1-\beta q^{-N})(\gamma\delta q)^{n}}{(1-\beta q^{2n-N})}\frac{\left(q% ,\beta q,\beta\gamma^{-1}q^{-N},\delta^{-1}q^{-N};q\right)_{n}}{\left(\beta q^% {-N},\beta\delta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad% \alpha q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\beta\gamma^{-1};q\right)_{N}}{% \left(\alpha\beta\gamma^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(% \beta^{-1}\gamma q^{-N})^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha% \beta q^{N+2},\alpha\beta\gamma^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,% \alpha\beta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\beta% \delta q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\delta^{-1};q\right)_{N}}{\left(% \alpha\delta^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(\delta q^{-N})% ^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta q^{N+2},\alpha\delta^% {-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,q^{-N};% q\right)_{n}}&<br/>\quad\textrm{if}\quad\gamma q=q^{-N}\end{array}\right.}}} } &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=% \lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( 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
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Racah polynomial : http://dlmf.nist.gov/18.28#E19
δ m , n subscript 𝛿 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.2 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:14.02:05&oldid=4229"