Formula:KLS:14.08:01: Difference between revisions

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Q n ( x ; a , b | q ) = ( a b ; q ) n a n \qHyperrphis 32 @ @ q - n , a e i θ , a e - i θ a b , 0 q q Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 𝑎 𝑏 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=% \frac{\left(ab;q\right)_{n}}{a^{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^% {\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab,0}{q}{q}}}}

Substitution(s)

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


Proof

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Symbols List

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
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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