Formula:6689

Digital Library of Mathematical Functions ID 18.28.E18

${\displaystyle{\displaystyle h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}\frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}e^{(n-2\ell)t}=e^{nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t}\right)={\mathrm{i}^{-n}}H_{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).}}$

Symbols List

• ${\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)}}$: continuous $$q$$ -Hermite polynomial
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}$: basic hypergeometric (or $$q$$ -hypergeometric) function
• ${\displaystyle{\displaystyle q}}$: real variable
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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