Formula:KLS:14.08:24: Difference between revisions

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Latest revision as of 08:36, 22 December 2019


w ~ ⁢ ( x ; a , b | q ) ⁢ Q n ⁡ ( x ; a , b | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n , b ⁢ q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑛 conditional 𝑏 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q)Q_{n}\!\left(x;a% ,b\,|\,q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{% q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n}|q)\right]% }}}

Substitution(s)

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

w ⁢ ( x ) := w ⁢ ( x ; a , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\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,a)h(x,b)% }}}} &
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
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
( 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
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
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 14.8 of KLS.

URL links

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