Formula:KLS:14.02:29

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\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , δ - 1 ⁢ q - x ⁢ γ ⁢ q ⁢ q ⁢ γ ⁢ δ ⁢ q x + 1 ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ α ⁢ q x + 1 , β ⁢ δ ⁢ q x + 1 ⁢ α ⁢ β ⁢ γ - 1 ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( α ⁢ q , β ⁢ δ ⁢ q ; q ) n ( α ⁢ β ⁢ γ - 1 ⁢ q , q ; q ) n ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝑞 𝑥 superscript 𝛿 1 superscript 𝑞 𝑥 𝛾 𝑞 𝑞 𝛾 𝛿 superscript 𝑞 𝑥 1 𝑡 \qHyperrphis 21 @ @ 𝛼 superscript 𝑞 𝑥 1 𝛽 𝛿 superscript 𝑞 𝑥 1 𝛼 𝛽 superscript 𝛾 1 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝛾 1 𝑞 𝑞 𝑞 𝑛 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\delta^{% -1}q^{-x}}{\gamma q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^% {x+1},\beta\delta q^{x+1}}{\alpha\beta\gamma^{-1}q}{q}{q^{-x}t}{}=\sum_{n=0}^{% N}\frac{\left(\alpha q,\beta\delta q;q\right)_{n}}{\left(\alpha\beta\gamma^{-1% }q,q;q\right)_{n}}R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)t% ^{n}{}}}}

Constraint(s)

if   α ⁢ q = q - N   or   β ⁢ δ ⁢ q = q - N formulae-sequence if 𝛼 𝑞 superscript 𝑞 𝑁 or 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad% \textrm{or}\quad\beta\delta q=q^{-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)}}} &
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

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Symbols List

& : logical and
ϕ s r subscript subscript italic-ϕ 𝑠 𝑟 {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}  : basic hypergeometric (or q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
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
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

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