Formula:KLS:14.02:05
Substitution(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}
&
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=\frac{\qPochhammer{\alpha^{-1}\beta^{-1}\gamma,\alpha^{-1}\delta,\beta^{-1},\gamma\delta q^2}{q}{\infty}} {\qPochhammer{\alpha^{-1}\beta^{-1}q^{-1},\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{\infty}} {}\frac{(1-\alpha\beta q)(\gamma\delta q)^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta\gamma^{-1}q,\alpha\delta^{-1}q,\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,\beta\delta q,\gamma q}{q}{n}} =\left\{\begin{array}{ll} \displaystyle\frac{\qPochhammer{\beta^{-1},\gamma\delta q^2}{q}{N}}{\qPochhammer{\beta^{-1}\gamma q,\delta q}{q}{N}} \frac{(1-\beta q^{-N})(\gamma\delta q)^n}{(1-\beta q^{2n-N})} \frac{\qPochhammer{q,\beta q,\beta\gamma^{-1}q^{-N},\delta^{-1}q^{-N}}{q}{n}}{\qPochhammer{\beta q^{-N},\beta\delta q,\gamma q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\alpha q=q^{-N}\\ \\ \displaystyle\frac{\qPochhammer{\alpha\beta q^2,\beta\gamma^{-1}}{q}{N}}{\qPochhammer{\alpha\beta\gamma^{-1}q,\beta q}{q}{N}} \frac{(1-\alpha\beta q)(\beta^{-1}\gamma q^{-N})^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\alpha\beta\gamma^{-1}q,\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,\gamma q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\beta\delta q=q^{-N}\\ \\ \displaystyle\frac{\qPochhammer{\alpha\beta q^2,\delta^{-1}}{q}{N}}{\qPochhammer{\alpha\delta^{-1}q,\beta q}{q}{N}} \frac{(1-\alpha\beta q)(\delta q^{-N})^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\alpha\delta^{-1}q,\beta q}{q}{n}} {\qPochhammer{\alpha q,\alpha\beta q,\beta\delta q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\gamma q=q^{-N} \end{array}\right.}}
} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x):=q^{-x}+\gamma\delta q^{x+1}}}
&
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}}}
&
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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}
: sum : http://drmf.wmflabs.org/wiki/Definition:sum
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (a;q)_n}}
: Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}}
-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R_{n}}}
: Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}}
-Racah polynomial : http://dlmf.nist.gov/18.28#E19
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_{m,n}}}
: Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{cos}}}
: cosine function : http://dlmf.nist.gov/4.14#E2
Bibliography
Equation in Section 14.2 of KLS.
URL links
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