Formula:KLS:14.10:76: Difference between revisions

From DRMF
Jump to navigation Jump to search
imported>SeedBot
DRMF
 
m Move page script moved page Formula:KLS:14.10:76 to F:KLS:14.10:76
 
(No difference)

Latest revision as of 08:37, 22 December 2019


C n ⁡ ( cos ⁡ θ ; β | q ) = ( β 2 ; q ) n ( q ; q ) n ⁢ β - 1 2 ⁢ n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - 1 2 ⁢ n , β ⁢ q 1 2 ⁢ n , β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ e - i ⁢ θ - β , β 1 2 ⁢ q 1 4 , - β 1 2 ⁢ q 1 4 ⁢ q 1 2 ⁢ q 1 2 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝜃 𝛽 𝑞 q-Pochhammer-symbol superscript 𝛽 2 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝛽 1 2 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 1 2 𝑛 𝛽 superscript 𝑞 1 2 𝑛 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 𝛽 superscript 𝛽 1 2 superscript 𝑞 1 4 superscript 𝛽 1 2 superscript 𝑞 1 4 superscript 𝑞 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\cos\theta;\beta\,|\,q% \right)=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}\beta^{-\frac% {1}{2}n}\qHyperrphis{4}{3}@@{q^{-\frac{1}{2}n},\beta q^{\frac{1}{2}n},\beta^{% \frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}{\mathrm{e}^{-% \mathrm{i}\theta}}}{-\beta,\beta^{\frac{1}{2}}q^{\frac{1}{4}},-\beta^{\frac{1}% {2}}q^{\frac{1}{4}}}{q^{\frac{1}{2}}}{q^{\frac{1}{2}}}}}}

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

C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2
( 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
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
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

Bibliography

Equation in Section 14.10 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:14.10:76&oldid=4863"