DLMF:18.28.E18 (Q5993)

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Language	Label	Description	Also known as
English	DLMF:18.28.E18	No description defined

Statements

Defining formula

{\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).}}

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DLMF:18.28.E18

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Symbols defined

continuous $$q^{-1}$$ -Hermite polynomial

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continuous $$q$$ -Hermite polynomial

Defining formula

{\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)}}

C18.S28.E16.m2abdec

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base of natural logarithm

Defining formula

{\displaystyle{\displaystyle\mathrm{e}}}

C4.S2.E11.m2aidec

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hyperbolic sine function

Defining formula

{\displaystyle{\displaystyle\sinh\NVar{z}}}

C4.S28.E1.m2adec

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Defining formula

{\displaystyle{\displaystyle\mathrm{i}}}

C1.S9.E1.m2ajdec

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Defining formula

{\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}

C17.S2.SS1.p1.m2amdec

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basic hypergeometric (or $$q$$ -hypergeometric) function

Defining formula

{\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)}}

C17.S4.E1.m2afdec

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Defining formula

{\displaystyle{\displaystyle q}}

C18.S1.XMD3.m1odec

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Defining formula

{\displaystyle{\displaystyle\ell}}

C18.S1.XMD4.m1edec

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Sitelinks

Wikipedia(0 entries)

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