Discrete q-Hermite II

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Discrete q-Hermite II

Basic hypergeometric representation

h ~ n ⁡ ( x ; q ) = i - n ⁢ V n ( - 1 ) ⁡ ( i ⁢ x ; q ) discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 imaginary-unit 𝑛 q-Al-Salam-Carlitz-II-polynomial-V 1 𝑛 imaginary-unit 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={% \mathrm{i}^{-n}}V^{(-1)}_{n}\!\left(\mathrm{i}x;q\right)}}} {\displaystyle \discrqHermiteII{n}@{x}{q}=\iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q} }
h ~ n ⁡ ( x ; q ) = i - n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q - n , i ⁢ x - q - q n discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 imaginary-unit 𝑛 superscript 𝑞 \binomial 𝑛 2 \qHyperrphis 20 @ @ superscript 𝑞 𝑛 imaginary-unit 𝑥 𝑞 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={% \mathrm{i}^{-n}}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\mathrm{i}x}% {-}{q}{-q^{n}}}}} {\displaystyle \discrqHermiteII{n}@{x}{q}=\iunit^{-n}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\iunit x}{-}{q}{-q^n} }
h ~ n ⁡ ( x ; q ) = x n ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - n + 1 ⁢ 0 ⁢ q 2 - q 2 x 2 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑥 𝑛 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 0 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)=x^{n% }\qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^{2}}{-\frac{q^{2}}{x^{2}}}}}} {\displaystyle \discrqHermiteII{n}@{x}{q}=x^n \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^2}{-\frac{q^2}{x^2}} }

Orthogonality relation(s)

∑ k = - ∞ ∞ [ h ~ m ⁡ ( c ⁢ q k ; q ) ⁢ h ~ n ⁡ ( c ⁢ q k ; q ) + h ~ m ⁡ ( - c ⁢ q k ; q ) ⁢ h ~ n ⁡ ( - c ⁢ q k ; q ) ] ⁢ w ⁢ ( c ⁢ q k ; q ) ⁢ q k = 2 ⁢ ( q 2 , - c 2 ⁢ q , - c - 2 ⁢ q ; q 2 ) ∞ ( q , - c 2 , - c - 2 ⁢ q 2 ; q 2 ) ∞ ⁢ ( q ; q ) n q n 2 ⁢ δ m , n superscript subscript 𝑘 delimited-[] discrete-q-Hermite-polynomial-II-h-tilde 𝑚 𝑐 superscript 𝑞 𝑘 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑐 superscript 𝑞 𝑘 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑚 𝑐 superscript 𝑞 𝑘 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑐 superscript 𝑞 𝑘 𝑞 𝑤 𝑐 superscript 𝑞 𝑘 𝑞 superscript 𝑞 𝑘 2 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑐 2 𝑞 superscript 𝑐 2 𝑞 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 superscript 𝑐 2 superscript 𝑐 2 superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 superscript 𝑛 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\left[% \tilde{h}_{m}\!\left(cq^{k};q\right)\tilde{h}_{n}\!\left(cq^{k};q\right)+% \tilde{h}_{m}\!\left(-cq^{k};q\right)\tilde{h}_{n}\!\left(-cq^{k};q\right)% \right]w(cq^{k};q)q^{k}{}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}\right)_{% \infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q% \right)_{n}}{q^{n^{2}}}\,\delta_{m,n}}}} {\displaystyle \sum_{k=-\infty}^{\infty}\left[\discrqHermiteII{m}@{cq^k}{q} \discrqHermiteII{n}@{cq^k}{q}+\discrqHermiteII{m}@{-cq^k}{q}\discrqHermiteII{n}@{-cq^k}{q}\right] w(cq^k;q)q^k {}=2\frac{\qPochhammer{q^2,-c^2q,-c^{-2}q}{q^2}{\infty}}{\qPochhammer{q,-c^2,-c^{-2}q^2}{q^2}{\infty}} \frac{\qPochhammer{q}{q}{n}}{q^{n^2}}\,\Kronecker{m}{n} }

Constraint(s): c > 0 𝑐 0 {\displaystyle{\displaystyle{\displaystyle c>0}}}


Substitution(s): w ⁢ ( x ; q ) = 1 ( i ⁢ x , - i ⁢ x ; q ) ∞ = 1 ( - x 2 ; q 2 ) ∞ 𝑤 𝑥 𝑞 1 q-Pochhammer-symbol imaginary-unit 𝑥 imaginary-unit 𝑥 𝑞 1 q-Pochhammer-symbol superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-% \mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}


∫ - ∞ ∞ h ~ m ⁡ ( x ; q ) ⁢ h ~ n ⁡ ( x ; q ) ( - x 2 ; q 2 ) ∞ ⁢ d q ⁢ x = ( q 2 , - q , - q ; q 2 ) ∞ ( q 3 , - q 2 , - q 2 ; q 2 ) ∞ ⁢ ( q ; q ) n q n 2 ⁢ δ m , n superscript subscript discrete-q-Hermite-polynomial-II-h-tilde 𝑚 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑥 2 superscript 𝑞 2 subscript 𝑑 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑞 2 𝑞 𝑞 superscript 𝑞 2 q-Pochhammer-symbol superscript 𝑞 3 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 superscript 𝑛 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\tilde{% h}_{m}\!\left(x;q\right)\tilde{h}_{n}\!\left(x;q\right)}{\left(-x^{2};q^{2}% \right)_{\infty}}\,d_{q}x=\frac{\left(q^{2},-q,-q;q^{2}\right)_{\infty}}{\left% (q^{3},-q^{2},-q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2% }}}\,\delta_{m,n}}}} {\displaystyle \int_{-\infty}^{\infty}\frac{\discrqHermiteII{m}@{x}{q}\discrqHermiteII{n}@{x}{q}}{\qPochhammer{-x^2}{q^2}{\infty}}\,d_qx =\frac{\qPochhammer{q^2,-q,-q}{q^2}{\infty}}{\qPochhammer{q^3,-q^2,-q^2}{q^2}{\infty}} \frac{\qPochhammer{q}{q}{n}}{q^{n^2}}\,\Kronecker{m}{n} }

Recurrence relation

x ⁢ h ~ n ⁡ ( x ; q ) = h ~ n + 1 ⁡ ( x ; q ) + q - 2 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ h ~ n - 1 ⁡ ( x ; q ) 𝑥 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑥 𝑞 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x\tilde{h}_{n}\!\left(x;q\right)=% \tilde{h}_{n+1}\!\left(x;q\right)+q^{-2n+1}(1-q^{n})\tilde{h}_{n-1}\!\left(x;q% \right)}}} {\displaystyle x\discrqHermiteII{n}@{x}{q}=\discrqHermiteII{n+1}@{x}{q}+q^{-2n+1}(1-q^n)\discrqHermiteII{n-1}@{x}{q} }

Monic recurrence relation

x ⁢ h ~ ^ n ⁡ ( x ) = h ~ ^ n + 1 ⁡ ( x ) + q - 2 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ h ~ ^ n - 1 ⁡ ( x ) 𝑥 discrete-q-Hermite-polynomial-II-monic-p 𝑛 𝑥 𝑞 discrete-q-Hermite-polynomial-II-monic-p 𝑛 1 𝑥 𝑞 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 discrete-q-Hermite-polynomial-II-monic-p 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{\tilde{h}}_{n}}\!\left(x% \right)={\widehat{\tilde{h}}_{n+1}}\!\left(x\right)+q^{-2n+1}(1-q^{n}){% \widehat{\tilde{h}}_{n-1}}\!\left(x\right)}}} {\displaystyle x\monicdiscrqHermiteII{n}@@{x}{q}=\monicdiscrqHermiteII{n+1}@@{x}{q}+q^{-2n+1}(1-q^n)\monicdiscrqHermiteII{n-1}@@{x}{q} }
h ~ n ⁡ ( x ; q ) = h ~ ^ n ⁡ ( x ) discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 discrete-q-Hermite-polynomial-II-monic-p 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={% \widehat{\tilde{h}}_{n}}\!\left(x\right)}}} {\displaystyle \discrqHermiteII{n}@{x}{q}=\monicdiscrqHermiteII{n}@@{x}{q} }

q-Difference equation

- ( 1 - q n ) ⁢ x 2 ⁢ h ~ n ⁡ ( x ; q ) = ( 1 + x 2 ) ⁢ h ~ n ⁡ ( q ⁢ x ; q ) - ( 1 + x 2 + q ) ⁢ h ~ n ⁡ ( x ; q ) + q ⁢ h ~ n ⁡ ( q - 1 ⁢ x ; q ) 1 superscript 𝑞 𝑛 superscript 𝑥 2 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 1 superscript 𝑥 2 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑞 𝑥 𝑞 1 superscript 𝑥 2 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 superscript 𝑞 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle-(1-q^{n})x^{2}\tilde{h}_{n}\!\left(% x;q\right){}=(1+x^{2})\tilde{h}_{n}\!\left(qx;q\right)-(1+x^{2}+q)\tilde{h}_{n% }\!\left(x;q\right)+q\tilde{h}_{n}\!\left(q^{-1}x;q\right)}}} {\displaystyle -(1-q^n)x^2\discrqHermiteII{n}@{x}{q} {}=(1+x^2)\discrqHermiteII{n}@{qx}{q}-(1+x^2+q)\discrqHermiteII{n}@{x}{q}+q\discrqHermiteII{n}@{q^{-1}x}{q} }

Forward shift operator

h ~ n ⁡ ( x ; q ) - h ~ n ⁡ ( q ⁢ x ; q ) = q - n + 1 ⁢ ( 1 - q n ) ⁢ x ⁢ h ~ n - 1 ⁡ ( q ⁢ x ; q ) discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑞 𝑥 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 𝑥 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)-% \tilde{h}_{n}\!\left(qx;q\right)=q^{-n+1}(1-q^{n})x\tilde{h}_{n-1}\!\left(qx;q% \right)}}} {\displaystyle \discrqHermiteII{n}@{x}{q}-\discrqHermiteII{n}@{qx}{q}=q^{-n+1}(1-q^n)x\discrqHermiteII{n-1}@{qx}{q} }
𝒟 q ⁢ h ~ n ⁡ ( x ; q ) = q - n + 1 ⁢ ( 1 - q n ) 1 - q ⁢ h ~ n - 1 ⁡ ( q ⁢ x ; q ) q-derivative 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\tilde{h}_{n}\!\left(% x;q\right)=\frac{q^{-n+1}(1-q^{n})}{1-q}\tilde{h}_{n-1}\!\left(qx;q\right)}}} {\displaystyle \qderiv{q}\discrqHermiteII{n}@{x}{q}=\frac{q^{-n+1}(1-q^n)}{1-q}\discrqHermiteII{n-1}@{qx}{q} }

Backward shift operator

h ~ n ⁡ ( x ; q ) - ( 1 + x 2 ) ⁢ h ~ n ⁡ ( q ⁢ x ; q ) = - q n ⁢ x ⁢ h ~ n + 1 ⁡ ( x ; q ) discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 1 superscript 𝑥 2 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑞 𝑥 𝑞 superscript 𝑞 𝑛 𝑥 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)-(1+x% ^{2})\tilde{h}_{n}\!\left(qx;q\right)=-q^{n}x\tilde{h}_{n+1}\!\left(x;q\right)% }}} {\displaystyle \discrqHermiteII{n}@{x}{q}-(1+x^2)\discrqHermiteII{n}@{qx}{q}=-q^nx\discrqHermiteII{n+1}@{x}{q} }
𝒟 q ⁡ [ w ⁢ ( x ; q ) ⁢ h ~ n ⁡ ( x ; q ) ] = - q n 1 - q ⁢ w ⁢ ( x ; q ) ⁢ h ~ n + 1 ⁡ ( x ; q ) q-derivative 𝑞 𝑤 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑞 𝑛 1 𝑞 𝑤 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left[w(x;q)\tilde{h}% _{n}\!\left(x;q\right)\right]=-\frac{q^{n}}{1-q}w(x;q)\tilde{h}_{n+1}\!\left(x% ;q\right)}}} {\displaystyle \qderiv{q}\left[w(x;q)\discrqHermiteII{n}@{x}{q}\right] =-\frac{q^n}{1-q}w(x;q)\discrqHermiteII{n+1}@{x}{q} }

Substitution(s): w ⁢ ( x ; q ) = 1 ( i ⁢ x , - i ⁢ x ; q ) ∞ = 1 ( - x 2 ; q 2 ) ∞ 𝑤 𝑥 𝑞 1 q-Pochhammer-symbol imaginary-unit 𝑥 imaginary-unit 𝑥 𝑞 1 q-Pochhammer-symbol superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-% \mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}


Rodrigues-type formula

w ⁢ ( x ; q ) ⁢ h ~ n ⁡ ( x ; q ) = ( q - 1 ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ ( 𝒟 q ) n ⁢ [ w ⁢ ( x ; q ) ] 𝑤 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑞 1 𝑛 superscript 𝑞 \binomial 𝑛 2 superscript q-derivative 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;q)\tilde{h}_{n}\!\left(x;q% \right)=(q-1)^{n}q^{-\binomial{n}{2}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x% ;q)\right]}}} {\displaystyle w(x;q)\discrqHermiteII{n}@{x}{q}=(q-1)^nq^{-\binomial{n}{2}} \left(\qderiv{q}\right)^n\left[w(x;q)\right] }

Substitution(s): w ⁢ ( x ; q ) = 1 ( i ⁢ x , - i ⁢ x ; q ) ∞ = 1 ( - x 2 ; q 2 ) ∞ 𝑤 𝑥 𝑞 1 q-Pochhammer-symbol imaginary-unit 𝑥 imaginary-unit 𝑥 𝑞 1 q-Pochhammer-symbol superscript 𝑥 2 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-% \mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}


Generating functions

( - x ⁢ t ; q ) ∞ ( - t 2 ; q 2 ) ∞ = ∑ n = 0 ∞ q \binomial ⁢ n ⁢ 2 ( q ; q ) n ⁢ h ~ n ⁡ ( x ; q ) ⁢ t n q-Pochhammer-symbol 𝑥 𝑡 𝑞 q-Pochhammer-symbol superscript 𝑡 2 superscript 𝑞 2 superscript subscript 𝑛 0 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(-xt;q\right)_{\infty}}{% \left(-t^{2};q^{2}\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2% }}}{\left(q;q\right)_{n}}\tilde{h}_{n}\!\left(x;q\right)t^{n}}}} {\displaystyle \frac{\qPochhammer{-xt}{q}{\infty}}{\qPochhammer{-t^2}{q^2}{\infty}}=\sum_{n=0}^{\infty} \frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\discrqHermiteII{n}@{x}{q}t^n }
( - i ⁢ t ; q ) ∞ ⋅ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ i ⁢ x - i ⁢ t ⁢ q ⁢ i ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q n ⁢ ( n - 1 ) ( q ; q ) n ⁢ h ~ n ⁡ ( x ; q ) ⁢ t n ⋅ q-Pochhammer-symbol imaginary-unit 𝑡 𝑞 \qHyperrphis 11 @ @ imaginary-unit 𝑥 imaginary-unit 𝑡 𝑞 imaginary-unit 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 𝑛 𝑛 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(-\mathrm{i}t;q\right)_{\infty}% \cdot\qHyperrphis{1}{1}@@{\mathrm{i}x}{-\mathrm{i}t}{q}{\mathrm{i}t}=\sum_{n=0% }^{\infty}\frac{(-1)^{n}q^{n(n-1)}}{\left(q;q\right)_{n}}\tilde{h}_{n}\!\left(% x;q\right)t^{n}}}} {\displaystyle \qPochhammer{-\iunit t}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{\iunit x}{-\iunit t}{q}{\iunit t}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{n(n-1)}}{\qPochhammer{q}{q}{n}}\discrqHermiteII{n}@{x}{q}t^n }

Limit relations

Al-Salam-Carlitz II polynomial to Discrete q-Hermite II polynomial

i - n ⁢ V n ( - 1 ) ⁡ ( i ⁢ x ; q ) = h ~ n ⁡ ( x ; q ) imaginary-unit 𝑛 q-Al-Salam-Carlitz-II-polynomial-V 1 𝑛 imaginary-unit 𝑥 𝑞 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle{\mathrm{i}^{-n}}V^{(-1)}_{n}\!\left% (\mathrm{i}x;q\right)=\tilde{h}_{n}\!\left(x;q\right)}}} {\displaystyle \iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q}=\discrqHermiteII{n}@{x}{q} }

Discrete q-Hermite II polynomial to Hermite polynomial

lim q → 1 ⁡ h ~ n ⁡ ( x ⁢ 1 - q 2 ; q ) ( 1 - q 2 ) n 2 = H n ⁡ ( x ) 2 n subscript → 𝑞 1 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 1 superscript 𝑞 2 𝑞 superscript 1 superscript 𝑞 2 𝑛 2 Hermite-polynomial-H 𝑛 𝑥 superscript 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{\tilde{h}% _{n}\!\left(x\sqrt{1-q^{2}};q\right)}{(1-q^{2})^{\frac{n}{2}}}=\frac{H_{n}% \left(x\right)}{2^{n}}}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\discrqHermiteII{n}@{x\sqrt{1-q^2}}{q}} {(1-q^2)^{\frac{n}{2}}}=\frac{\Hermite{n}@{x}}{2^n} }

Remark

h ~ n ⁡ ( x ; q - 1 ) = i - n ⁢ h n ⁡ ( i ⁢ x ; q ) discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 superscript 𝑞 1 imaginary-unit 𝑛 discrete-q-Hermite-polynomial-h-I 𝑛 imaginary-unit 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q^{-1}\right)% ={\mathrm{i}^{-n}}h_{n}\!\left(\mathrm{i}x;q\right)}}} {\displaystyle \discrqHermiteII{n}@{x}{q^{-1}}=\iunit^{-n}\discrqHermiteI{n}@{\iunit x}{q} }

Koornwinder Addendum: Discrete q-Hermite II

Basic hypergeometric representation

h ~ n ⁡ ( x ; q ) = x n ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q - n + 1 ⁢ 0 ⁢ q 2 - q 2 ⁢ x - 2 discrete-q-Hermite-polynomial-II-h-tilde 𝑛 𝑥 𝑞 superscript 𝑥 𝑛 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 0 superscript 𝑞 2 superscript 𝑞 2 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)=x^{n% }\qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^{2}}{-q^{2}x^{-2}}}}} {\displaystyle \discrqHermiteII{n}@{x}{q}=x^n \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^2}{-q^2 x^{-2}} }
p n ⁡ ( 0 ; a , b ; q ) = 1 little-q-Jacobi-polynomial-p 𝑛 0 𝑎 𝑏 𝑞 1 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(0;a,b;q\right)=1}}} {\displaystyle \littleqJacobi{n}@{0}{a}{b}{q}=1 }
p n ⁡ ( q - 1 ⁢ b - 1 ; a , b ; q ) = ( - q ⁢ b ) - n ⁢ q - 1 2 ⁢ n ⁢ ( n - 1 ) ⁢ ( q ⁢ b ; q ) n ( q ⁢ a ; q ) n little-q-Jacobi-polynomial-p 𝑛 superscript 𝑞 1 superscript 𝑏 1 𝑎 𝑏 𝑞 superscript 𝑞 𝑏 𝑛 superscript 𝑞 1 2 𝑛 𝑛 1 q-Pochhammer-symbol 𝑞 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(q^{-1}b^{-1};a,b;q% \right)=(-qb)^{-n}q^{-\frac{1}{2}n(n-1)}\frac{\left(qb;q\right)_{n}}{\left(qa;% q\right)_{n}}}}} {\displaystyle \littleqJacobi{n}@{q^{-1}b^{-1}}{a}{b}{q}=(-qb)^{-n} q^{-\frac12 n(n-1)} \frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}} }
p n ⁡ ( 1 ; a , b ; q ) = ( - a ) n ⁢ q 1 2 ⁢ n ⁢ ( n + 1 ) ⁢ ( q ⁢ b ; q ) n ( q ⁢ a ; q ) n little-q-Jacobi-polynomial-p 𝑛 1 𝑎 𝑏 𝑞 superscript 𝑎 𝑛 superscript 𝑞 1 2 𝑛 𝑛 1 q-Pochhammer-symbol 𝑞 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(1;a,b;q\right)=(-a)^{n% }q^{\frac{1}{2}n(n+1)}\frac{\left(qb;q\right)_{n}}{\left(qa;q\right)_{n}}}}} {\displaystyle \littleqJacobi{n}@{1}{a}{b}{q}=(-a)^n q^{\frac12 n(n+1)} \frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}} }
P n ( α , β ) ⁡ ( - x | q ) = ( - 1 ) n ⁢ q 1 2 ⁢ ( α - β ) ⁢ n ⁢ P n ( β , α ) ⁡ ( x | q ) continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 superscript 1 𝑛 superscript 𝑞 1 2 𝛼 𝛽 𝑛 continuous-q-Jacobi-polynomial-P 𝛽 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(-x|q% \right)=(-1)^{n}q^{\frac{1}{2}(\alpha-\beta)n}P^{(\beta,\alpha)}_{n}\!\left(x|% q\right)}}} {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{-x }{ q}=(-1)^n q^{\frac12(\alpha-\beta)n} \ctsqJacobi{\beta}{\alpha}{n}@{x }{ q} }

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