Continuous big q-Hermite

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Continuous big q-Hermite

Basic hypergeometric representation

H n ⁡ ( x ; a | q ) = a - n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ ⁢ 0 , 0 ⁢ q ⁢ q continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 0 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;a|q\right)=a^{-n}\,% \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-% \mathrm{i}\theta}}}{0,0}{q}{q}}}} {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{0,0}{q}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


H n ⁡ ( x ; a | q ) = e i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ - q ⁢ q n ⁢ e - 2 ⁢ i ⁢ θ continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 imaginary-unit 𝑛 𝜃 \qHyperrphis 20 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 𝑞 superscript 𝑞 𝑛 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;a|q\right)={\mathrm{% e}^{\mathrm{i}n\theta}}\,\qHyperrphis{2}{0}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}% \theta}}}{-}{q}{q^{n}{\mathrm{e}^{-2\mathrm{i}\theta}}}}}} {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=\expe^{\iunit n\theta}\,\qHyperrphis{2}{0}@@{q^{-n},a\expe^{\iunit\theta}}{-}{q}{q^n\expe^{-2\iunit\theta}} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ H m ⁡ ( x ; a | q ) ⁢ H n ⁡ ( x ; a | q ) ⁢ 𝑑 x = δ m , n ( q n + 1 ; q ) ∞ 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-big-q-Hermite-polynomial-H 𝑚 𝑥 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 differential-d 𝑥 Kronecker-delta 𝑚 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}H_{m}\!\left(x;a|q\right)H_{n}\!\left(x;a|q\right)\,dx=\frac% {\,\delta_{m,n}}{\left(q^{n+1};q\right)_{\infty}}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsbigqHermite{m}@{x}{a}{q}\ctsbigqHermite{n}@{x}{a}{q}\,dx =\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1}}{q}{\infty}} }

Substitution(s): w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &

h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ H m ⁡ ( x ; a | q ) ⁢ H n ⁡ ( x ; a | q ) ⁢ 𝑑 x + ∑ k 1 < a ⁢ q k ≤ a w k ⁢ H m ⁡ ( x k ; a | q ) ⁢ H n ⁡ ( x k ; a | q ) = δ m , n ( q n + 1 ; q ) ∞ 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-big-q-Hermite-polynomial-H 𝑚 𝑥 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 differential-d 𝑥 subscript 𝑘 1 𝑎 superscript 𝑞 𝑘 𝑎 subscript 𝑤 𝑘 continuous-big-q-Hermite-polynomial-H 𝑚 subscript 𝑥 𝑘 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 subscript 𝑥 𝑘 𝑎 𝑞 Kronecker-delta 𝑚 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}H_{m}\!\left(x;a|q\right)H_{n}\!\left(x;a|q\right)\,dx{}+% \sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}H_{m}\!\left(x_{k};a|q\right)H_{n}% \!\left(x_{k};a|q\right)=\frac{\,\delta_{m,n}}{\left(q^{n+1};q\right)_{\infty}% }}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsbigqHermite{m}@{x}{a}{q}\ctsbigqHermite{n}@{x}{a}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\ \scriptstyle 1

Substitution(s): w k = ( a - 2 ; q ) ∞ ( q ; q ) ∞ ⁢ ( 1 - a 2 ⁢ q 2 ⁢ k ) ⁢ ( a 2 ; q ) k ( 1 - a 2 ) ⁢ ( q ; q ) k ⁢ q - 3 2 ⁢ k 2 - 1 2 ⁢ k ⁢ ( - 1 a 4 ) k subscript 𝑤 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑞 q-Pochhammer-symbol 𝑞 𝑞 1 superscript 𝑎 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑞 𝑘 1 superscript 𝑎 2 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 𝑞 3 2 superscript 𝑘 2 1 2 𝑘 superscript 1 superscript 𝑎 4 𝑘 {\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{% \infty}}{\left(q;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})\left(a^{2};q\right)_{% k}}{(1-a^{2})\left(q;q\right)_{k}}q^{-\frac{3}{2}k^{2}-\frac{1}{2}k}\left(-% \frac{1}{a^{4}}\right)^{k}}}} &

x k = a ⁢ q k + ( a ⁢ q k ) - 1 2 subscript 𝑥 𝑘 𝑎 superscript 𝑞 𝑘 superscript 𝑎 superscript 𝑞 𝑘 1 2 {\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}% \right)^{-1}}{2}}}} &
w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


Recurrence relation

2 ⁢ x ⁢ H n ⁡ ( x ; a | q ) = H n + 1 ⁡ ( x ; a | q ) + a ⁢ q n ⁢ H n ⁡ ( x ; a | q ) + ( 1 - q n ) ⁢ H n - 1 ⁡ ( x ; a | q ) 2 𝑥 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xH_{n}\!\left(x;a|q\right)=H_{n+1}% \!\left(x;a|q\right)+aq^{n}H_{n}\!\left(x;a|q\right)+(1-q^{n})H_{n-1}\!\left(x% ;a|q\right)}}} {\displaystyle 2x\ctsbigqHermite{n}@{x}{a}{q}=\ctsbigqHermite{n+1}@{x}{a}{q}+aq^n\ctsbigqHermite{n}@{x}{a}{q}+(1-q^n)\ctsbigqHermite{n-1}@{x}{a}{q} }

Monic recurrence relation

x ⁢ H ^ n ⁡ ( x ) = H ^ n + 1 ⁡ ( x ) + 1 2 ⁢ a ⁢ q n ⁢ H ^ n ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ H ^ n - 1 ⁡ ( x ) 𝑥 continuous-big-q-Hermite-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 continuous-big-q-Hermite-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 1 2 𝑎 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 1 4 1 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{H}}_{n}\!\left(x\right)=% {\widehat{H}}_{n+1}\!\left(x\right)+\frac{1}{2}aq^{n}{\widehat{H}}_{n}\!\left(% x\right)+\frac{1}{4}(1-q^{n}){\widehat{H}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsbigqHermite{n}@@{x}{a}{q}=\monicctsbigqHermite{n+1}@@{x}{a}{q}+\frac{1}{2}aq^n\monicctsbigqHermite{n}@@{x}{a}{q}+\frac{1}{4}(1-q^n)\monicctsbigqHermite{n-1}@@{x}{a}{q} }
H n ⁡ ( x ; a | q ) = 2 n ⁢ H ^ n ⁡ ( x ) continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 2 𝑛 continuous-big-q-Hermite-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;a|q\right)=2^{n}{% \widehat{H}}_{n}\!\left(x\right)}}} {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=2^n\monicctsbigqHermite{n}@@{x}{a}{q} }

q-Difference equations

( 1 - q ) 2 ⁢ D q ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 | q ) ⁢ D q ⁢ y ⁢ ( x ) ] + 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ w ~ ⁢ ( x ; a | q ) ⁢ y ⁢ ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq% ^{\frac{1}{2}}|q)D_{q}y(x)\right]+4q^{-n+1}(1-q^{n}){\tilde{w}}(x;a|q)y(x)=0}}} {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}}|q)D_qy(x)\right] +4q^{-n+1}(1-q^n){\tilde w}(x;a|q)y(x)=0 }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

y ⁢ ( x ) = H n ⁡ ( x ; a | q ) 𝑦 𝑥 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=H_{n}\!\left(x;a|q\right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


\LegendrePoly ⁢ n ⁢ @ ⁢ z := a - n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ z , a ⁢ z - 1 ⁢ 0 , 0 ⁢ q ⁢ q assign \LegendrePoly 𝑛 @ 𝑧 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 𝑧 𝑎 superscript 𝑧 1 0 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{z}:=a^{-n}\,% \qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{0,0}{q}{q}}}} {\displaystyle \LegendrePoly{n}@{z}:=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{0,0}{q}{q} }
q - n ⁢ ( 1 - q n ) ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ z = A ⁢ ( z ) ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ q ⁢ z - [ A ⁢ ( z ) + A ⁢ ( z - 1 ) ] ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ z + A ⁢ ( z - 1 ) ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ q - 1 ⁢ z superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 \LegendrePoly 𝑛 @ 𝑧 𝐴 𝑧 \LegendrePoly 𝑛 @ 𝑞 𝑧 delimited-[] 𝐴 𝑧 𝐴 superscript 𝑧 1 \LegendrePoly 𝑛 @ 𝑧 𝐴 superscript 𝑧 1 \LegendrePoly 𝑛 @ superscript 𝑞 1 𝑧 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})\LegendrePoly{n}@{z}% =A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z}{}+A% (z^{-1})\LegendrePoly{n}@{q^{-1}z}}}} {\displaystyle q^{-n}(1-q^n)\LegendrePoly{n}@{z}=A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z} {}+A(z^{-1})\LegendrePoly{n}@{q^{-1}z} }

Substitution(s): A ⁢ ( z ) = ( 1 - a ⁢ z ) ( 1 - z 2 ) ⁢ ( 1 - q ⁢ z 2 ) 𝐴 𝑧 1 𝑎 𝑧 1 superscript 𝑧 2 1 𝑞 superscript 𝑧 2 {\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)}{(1-z^{2})(1-qz^{% 2})}}}}


Forward shift operator

δ q ⁢ H n ⁡ ( x ; a | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - q n ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ H n - 1 ⁡ ( x ; a ⁢ q 1 2 | q ) subscript 𝛿 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 imaginary-unit 𝜃 imaginary-unit 𝜃 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}H_{n}\!\left(x;a|q\right)=% -q^{-\frac{1}{2}n}(1-q^{n})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-% \mathrm{i}\theta}})H_{n-1}\!\left(x;aq^{\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q\ctsbigqHermite{n}@{x}{a}{q}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) \ctsbigqHermite{n-1}@{x}{aq^{\frac{1}{2}}}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


D q ⁢ H n ⁡ ( x ; a | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) ⁢ ( 1 - q n ) 1 - q ⁢ H n - 1 ⁡ ( x ; a ⁢ q 1 2 | q ) subscript 𝐷 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}H_{n}\!\left(x;a|q\right)=% \frac{2q^{-\frac{1}{2}(n-1)}(1-q^{n})}{1-q}H_{n-1}\!\left(x;aq^{\frac{1}{2}}|q% \right)}}} {\displaystyle D_q\ctsbigqHermite{n}@{x}{a}{q}=\frac{2q^{-\frac{1}{2}(n-1)}(1-q^n)}{1-q} \ctsbigqHermite{n-1}@{x}{aq^{\frac{1}{2}}}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; a | q ) ⁢ H n ⁡ ( x ; a | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 | q ) ⁢ H n + 1 ⁡ ( x ; a ⁢ q - 1 2 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a|q)H_% {n}\!\left(x;a|q\right)\right]{}=q^{-\frac{1}{2}(n+1)}({\mathrm{e}^{\mathrm{i}% \theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){}{\tilde{w}}(x;aq^{-\frac{1}{2}}|q)% H_{n+1}\!\left(x;aq^{-\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;aq^{-\frac{1}{2}}|q)\ctsbigqHermite{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


D q ⁢ [ w ~ ⁢ ( x ; a | q ) ⁢ H n ⁡ ( x ; a | q ) ] = - 2 ⁢ q - 1 2 ⁢ n 1 - q ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 | q ) ⁢ H n + 1 ⁡ ( x ; a ⁢ q - 1 2 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 2 superscript 𝑞 1 2 𝑛 1 𝑞 ~ 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a|q)H_{n}% \!\left(x;a|q\right)\right]=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}}(x;aq^{-% \frac{1}{2}}|q)H_{n+1}\!\left(x;aq^{-\frac{1}{2}}|q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}\right]= -\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}}|q)\ctsbigqHermite{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a | q ) := w ⁢ ( x ; a | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝑎 𝑞 𝑤 𝑥 conditional 𝑎 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a|q):=\frac{w(x;a|q)}{% \sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


Rodrigues-type formula

w ⁢ ( x ; a | q ) ⁢ H n ⁡ ( x ; a | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ⁢ ( x ; a ⁢ q 1 2 ⁢ n | q ) ] 𝑤 𝑥 conditional 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] 𝑤 𝑥 conditional 𝑎 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;a|q)H_{n}\!\left(x;a|q\right)=% \left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{q}\right)^{n}\left% [w(x;aq^{\frac{1}{2}n}|q)\right]}}} {\displaystyle w(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)} \left(D_q\right)^n\left[w(x;aq^{\frac{1}{2}n}|q)\right] }

Substitution(s): w ⁢ ( x ) := w ⁢ ( x ; a | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝑎 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a|q)=\left|\frac{\left({% \mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm% {i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}} &

h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


Generating functions

( a ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t , e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ H n ⁡ ( x ; a | q ) ( q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑎 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(at;q\right)_{\infty}}{% \left({\mathrm{e}^{\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm{i}\theta}}t;q% \right)_{\infty}}=\sum_{n=0}^{\infty}\frac{H_{n}\!\left(x;a|q\right)}{\left(q;% q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{at}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\ctsbigqHermite{n}@{x}{a}{q}}{\qPochhammer{q}{q}{n}}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( e i ⁢ θ ⁢ t ; q ) ∞ ⋅ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q ; q ) n ⁢ H n ⁡ ( x ; a | q ) ⁢ t n ⋅ q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 11 @ @ 𝑎 imaginary-unit 𝜃 imaginary-unit 𝜃 𝑡 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left({\mathrm{e}^{\mathrm{i}\theta}% }t;q\right)_{\infty}\cdot\qHyperrphis{1}{1}@@{a{\mathrm{e}^{\mathrm{i}\theta}}% }{{\mathrm{e}^{\mathrm{i}\theta}}t}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}{}=% \sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}H_{% n}\!\left(x;a|q\right)t^{n}}}} {\displaystyle \qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{a\expe^{\iunit\theta}}{\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\ctsbigqHermite{n}@{x}{a}{q}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( γ ⁢ e i ⁢ θ ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , a ⁢ e i ⁢ θ ⁢ γ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( γ ; q ) n ( q ; q ) n ⁢ H n ⁡ ( x ; a | q ) ⁢ t n q-Pochhammer-symbol 𝛾 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝛾 𝑎 imaginary-unit 𝜃 𝛾 imaginary-unit 𝜃 𝑡 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma{\mathrm{e}^{% \mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;% q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{\gamma,a{\mathrm{e}^{\mathrm{i}\theta% }}}{\gamma{\mathrm{e}^{\mathrm{i}\theta}}t}{q}{{\mathrm{e}^{-\mathrm{i}\theta}% }t}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q\right)_{n}}{\left(q;q\right)_{n}% }H_{n}\!\left(x;a|q\right)t^{n}}}} {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{\gamma,a\expe^{\iunit\theta}}{\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q}{q}{n}}\ctsbigqHermite{n}@{x}{a}{q}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Al-Salam-Chihara polynomial to Continuous big q-Hermite polynomial

Q n ⁡ ( x ; a , 0 | q ) = H n ⁡ ( x ; a | q ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 0 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,0\,|\,q\right)=H_{% n}\!\left(x;a|q\right)}}} {\displaystyle \AlSalamChihara{n}@{x}{a}{0}{q}=\ctsbigqHermite{n}@{x}{a}{q} }

Continuous big q-Hermite polynomial to Continuous q-Hermite polynomial

H n ⁡ ( x ; 0 | q ) = H n ⁡ ( x | q ) continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 0 𝑞 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;0|q\right)=H_{n}\!% \left(x\,|\,q\right)}}} {\displaystyle \ctsbigqHermite{n}@{x}{0}{q}=\ctsqHermite{n}@{x}{q} }

Continuous big q-Hermite polynomial to Hermite polynomial

lim q → 1 ⁡ H n ⁡ ( x ⁢ 1 2 ⁢ ( 1 - q ) ; 0 | q ) ( 1 - q 2 ) n 2 = H n ⁡ ( x ) subscript → 𝑞 1 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 1 2 1 𝑞 0 𝑞 superscript 1 𝑞 2 𝑛 2 Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{H_{n}\!% \left(x\sqrt{\frac{1}{2}(1-q)};0|q\right)}{\left(\frac{1-q}{2}\right)^{\frac{n% }{2}}}=H_{n}\left(x\right)}}} {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{0}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x} }
lim q → 1 ⁡ H n ⁡ ( x ⁢ 1 2 ⁢ ( 1 - q ) ; a ⁢ 2 ⁢ ( 1 - q ) | q ) ( 1 - q 2 ) n 2 = H n ⁡ ( x - a ) subscript → 𝑞 1 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 1 2 1 𝑞 𝑎 2 1 𝑞 𝑞 superscript 1 𝑞 2 𝑛 2 Hermite-polynomial-H 𝑛 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{H_{n}\!% \left(x\sqrt{\frac{1}{2}(1-q)};a\sqrt{2(1-q)}|q\right)}{\left(\frac{1-q}{2}% \right)^{\frac{n}{2}}}=H_{n}\left(x-a\right)}}} {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{a\sqrt{2(1-q)}}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x-a} }

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