Continuous q-Hermite

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Continuous q-Hermite

Basic hypergeometric representation

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

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


Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x | q ) 1 - x 2 ⁢ H m ⁡ ( x | q ) ⁢ H n ⁡ ( x | q ) ⁢ 𝑑 x = δ m , n ( q n + 1 ; q ) ∞ 1 2 superscript subscript 1 1 𝑤 conditional 𝑥 𝑞 1 superscript 𝑥 2 continuous-q-Hermite-polynomial-H 𝑚 𝑥 𝑞 continuous-q-Hermite-polynomial-H 𝑛 𝑥 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% |q)}{\sqrt{1-x^{2}}}H_{m}\!\left(x\,|\,q\right)H_{n}\!\left(x\,|\,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|q)}{\sqrt{1-x^2}}\ctsqHermite{m}@{x}{q}\ctsqHermite{n}@{x}{q}\,dx= \frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1}}{q}{\infty}} }

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

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

Monic recurrence relation

x ⁢ H ^ n ⁡ ( x ) = H ^ n + 1 ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ H ^ n - 1 ⁡ ( x ) 𝑥 continuous-q-Hermite-polynomial-monic-p 𝑛 𝑥 𝑞 continuous-q-Hermite-polynomial-monic-p 𝑛 1 𝑥 𝑞 1 4 1 superscript 𝑞 𝑛 continuous-q-Hermite-polynomial-monic-p 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{H}}_{n}\!\left(x\right)=% {\widehat{H}}_{n+1}\!\left(x\right)+\frac{1}{4}(1-q^{n}){\widehat{H}}_{n-1}\!% \left(x\right)}}} {\displaystyle x\monicctsqHermite{n}@@{x}{q}=\monicctsqHermite{n+1}@@{x}{q}+\frac{1}{4}(1-q^n)\monicctsqHermite{n-1}@@{x}{q} }
H n ⁡ ( x | q ) = 2 n ⁢ H ^ n ⁡ ( x ) continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 superscript 2 𝑛 continuous-q-Hermite-polynomial-monic-p 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x\,|\,q\right)=2^{n}{% \widehat{H}}_{n}\!\left(x\right)}}} {\displaystyle \ctsqHermite{n}@{x}{q}=2^n\monicctsqHermite{n}@@{x}{q} }

q-Difference equation

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

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

y ⁢ ( x ) = H n ⁡ ( x | q ) 𝑦 𝑥 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=H_{n}\!\left(x\,|\,q\right)}}} &
w ⁢ ( x | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) 𝑤 conditional 𝑥 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x|q)=\left|\left({\mathrm{e}^{2% \mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}}} &
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}}}


Forward shift operator

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

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


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

Backward shift operator

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

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

w ⁢ ( x | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) 𝑤 conditional 𝑥 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x|q)=\left|\left({\mathrm{e}^{2% \mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}}} &
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 | q ) ⁢ H n ⁡ ( x | q ) ] = - 2 ⁢ q - 1 2 ⁢ n 1 - q ⁢ w ~ ⁢ ( x | q ) ⁢ H n + 1 ⁡ ( x | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 conditional 𝑥 𝑞 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 2 superscript 𝑞 1 2 𝑛 1 𝑞 ~ 𝑤 conditional 𝑥 𝑞 continuous-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x|q)H_{n}\!% \left(x\,|\,q\right)\right]=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}}(x|q)H_{% n+1}\!\left(x\,|\,q\right)}}} {\displaystyle D_q\left[{\tilde w}(x|q)\ctsqHermite{n}@{x}{q}\right]= -\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x|q)\ctsqHermite{n+1}@{x}{q} }

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

w ⁢ ( x | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) 𝑤 conditional 𝑥 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x|q)=\left|\left({\mathrm{e}^{2% \mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-q^{\frac{1}{2}})}}} &
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 | q ) ⁢ H n ⁡ ( x | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x | q ) ] ~ 𝑤 conditional 𝑥 𝑞 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 conditional 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x|q)H_{n}\!\left(x\,|\,q% \right)=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{q}\right)^% {n}\left[{\tilde{w}}(x|q)\right]}}} {\displaystyle {\tilde w}(x|q)\ctsqHermite{n}@{x}{q}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)} \left(D_q\right)^n\left[{\tilde w}(x|q)\right] }

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

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

1 | ( e i ⁢ θ ⁢ t ; q ) ∞ | 2 = 1 ( e i ⁢ θ ⁢ t , e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ H n ⁡ ( x | q ) ( q ; q ) n ⁢ t n 1 superscript q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 2 1 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left|\left({\mathrm{e}^{% \mathrm{i}\theta}}t;q\right)_{\infty}\right|^{2}}=\frac{1}{\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\,|\,q\right)}{\left(q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{1}{\left|\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\right|^2}= \frac{1}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}}= \sum_{n=0}^{\infty}\frac{\ctsqHermite{n}@{x}{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 ⁢ @ ⁢ @ ⁢ 0 ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q ; q ) n ⁢ H n ⁡ ( x | q ) ⁢ t n ⋅ q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 11 @ @ 0 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-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left({\mathrm{e}^{\mathrm{i}\theta}% }t;q\right)_{\infty}\cdot\qHyperrphis{1}{1}@@{0}{{\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\,|\,q\right)t^{n}}}} {\displaystyle \qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{0}{\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\ctsqHermite{n}@{x}{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 , 0 ⁢ γ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( γ ; q ) n ( q ; q ) n ⁢ H n ⁡ ( x | q ) ⁢ t n q-Pochhammer-symbol 𝛾 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝛾 0 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-q-Hermite-polynomial-H 𝑛 𝑥 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,0}{\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\,|\,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,0}{\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q}{q}{n}}\ctsqHermite{n}@{x}{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

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 q-Laguerre polynomial to Continuous q-Hermite polynomial

lim α → ∞ ⁡ P n ( α ) ⁡ ( x | q ) q ( 1 2 ⁢ α + 1 4 ) ⁢ n = H n ⁡ ( x | q ) ( q ; q ) n subscript → 𝛼 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\frac{% P^{(\alpha)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}=% \frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \frac{\ctsqLaguerre{\alpha}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}} =\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}} }

Continuous q-Hermite polynomial to Hermite polynomial

lim q → 1 ⁡ H n ⁡ ( x ⁢ 1 2 ⁢ ( 1 - q ) | q ) ( 1 - q 2 ) n 2 = H n ⁡ ( x ) subscript → 𝑞 1 continuous-q-Hermite-polynomial-H 𝑛 𝑥 1 2 1 𝑞 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)}\,|\,q\right)}{\left(\frac{1-q}{2}\right)^{\frac% {n}{2}}}=H_{n}\left(x\right)}}} {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x} }

Remark

H n ⁡ ( x | q ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ⁢ ( q ; q ) n - k ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x\,|\,q\right)=\sum_{k% =0}^{n}\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}% {\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}} {\displaystyle \ctsqHermite{n}@{x}{q}=\sum_{k=0}^n\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}}\expe^{\iunit(n-2k)\theta} }

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


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