Formula:KLS:14.10:20: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


δ q ⁢ [ w ~ ⁢ ( x ; q α , q β | q ) ⁢ P n ( α , β ) ⁡ ( x | q ) ] = q - 1 2 ⁢ α - 1 4 ⁢ ( 1 - q n + 1 ) ⁢ ( 1 + q 1 2 ⁢ ( α + β - 1 ) ) ⁢ ( 1 + q 1 2 ⁢ ( α + β ) ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; q α - 1 , q β - 1 | q ) ⁢ P n + 1 ( α - 1 , β - 1 ) ⁢ ( x | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 1 2 𝛼 𝛽 1 1 superscript 𝑞 1 2 𝛼 𝛽 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 superscript 𝑞 𝛼 1 conditional superscript 𝑞 𝛽 1 𝑞 superscript subscript 𝑃 𝑛 1 𝛼 1 𝛽 1 conditional 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;q^{% \alpha},q^{\beta}|q)P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\right]{}=q^{-% \frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)})(1% +q^{\frac{1}{2}(\alpha+\beta)})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-% \mathrm{i}\theta}}){}{\tilde{w}}(x;q^{\alpha-1},q^{\beta-1}|q)P_{n+1}^{(\alpha% -1,\beta-1)}(x|q)}}}

Substitution(s)

w ~ ⁢ ( x ; q α , q β | q ) := w ⁢ ( x ; q α , q β | q ) 1 - x 2 assign ~ 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha},q^{\beta}|q% ):=\frac{w(x;q^{\alpha},q^{\beta}|q)}{\sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; q α , q β | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ , q 1 2 ⁢ α + 3 4 ⁢ e i ⁢ θ - q 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ , - q 1 2 ⁢ β + 3 4 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e i ⁢ θ , - e i ⁢ θ ; q 1 2 ) ∞ ( q 1 2 ⁢ α + 1 4 ⁢ e i ⁢ θ - q 1 2 ⁢ β + 1 4 ⁢ e i ⁢ θ ; q 1 2 ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , q 1 2 ⁢ α + 1 4 ) ⁢ h ⁢ ( x , q 1 2 ⁢ α + 3 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 1 4 ) ⁢ h ⁢ ( x , - q 1 2 ⁢ β + 3 4 ) assign 𝑤 𝑥 𝑤 𝑥 superscript 𝑞 𝛼 conditional superscript 𝑞 𝛽 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛼 3 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 3 4 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol imaginary-unit 𝜃 imaginary-unit 𝜃 superscript 𝑞 1 2 q-Pochhammer-symbol superscript 𝑞 1 2 𝛼 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 𝛽 1 4 imaginary-unit 𝜃 superscript 𝑞 1 2 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 𝛼 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛼 3 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 1 4 ℎ 𝑥 superscript 𝑞 1 2 𝛽 3 4 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=% \left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^% {\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}% \alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1% }{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm% {e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({% \mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}% }\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm% {i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^% {\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1% }{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}% {2}\beta+\frac{3}{4}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Proof

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Symbols List

& : logical and
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
( 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
Π Π {\displaystyle{\displaystyle{\displaystyle\Pi}}}  : product : http://drmf.wmflabs.org/wiki/Definition:prod
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.10 of KLS.

URL links

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