Al-Salam-Chihara

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Al-Salam-Chihara

Basic hypergeometric representation

Q n ⁡ ( x ; a , b | q ) = ( a ⁢ b ; q ) n a n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ ⁢ a ⁢ b , 0 ⁢ q ⁢ q Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 𝑎 𝑏 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=% \frac{\left(ab;q\right)_{n}}{a^{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^% {\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab,0}{q}{q}}}} {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\frac{\qPochhammer{ab}{q}{n}}{a^n}\ \qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,0}{q}{q} }

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


Q n ⁡ ( x ; a , b | q ) = ( a ⁢ e i ⁢ θ ; q ) n ⁢ e - i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , b ⁢ e - i ⁢ θ ⁢ a - 1 ⁢ q - n + 1 ⁢ e - i ⁢ θ ⁢ q ⁢ a - 1 ⁢ q ⁢ e i ⁢ θ Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑞 𝑛 imaginary-unit 𝑛 𝜃 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑏 imaginary-unit 𝜃 superscript 𝑎 1 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 𝑞 superscript 𝑎 1 𝑞 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=% \left(a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{n}{\mathrm{e}^{-\mathrm{i}n% \theta}}\,\qHyperrphis{2}{1}@@{q^{-n},b{\mathrm{e}^{-\mathrm{i}\theta}}}{a^{-1% }q^{-n+1}{\mathrm{e}^{-\mathrm{i}\theta}}}{q}{a^{-1}q{\mathrm{e}^{\mathrm{i}% \theta}}}}}} {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\qPochhammer{a\expe^{\iunit\theta}}{q}{n}\expe^{-\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},b\expe^{-\iunit\theta}}{a^{-1}q^{-n+1}\expe^{-\iunit\theta}}{q}{a^{-1}q\expe^{\iunit\theta}} }
Q n ⁡ ( x ; a , b | q ) = ( b ⁢ e - i ⁢ θ ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ ⁢ b - 1 ⁢ q - n + 1 ⁢ e i ⁢ θ ⁢ q ⁢ b - 1 ⁢ q ⁢ e - i ⁢ θ Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑏 imaginary-unit 𝜃 𝑞 𝑛 imaginary-unit 𝑛 𝜃 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 superscript 𝑏 1 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 𝑞 superscript 𝑏 1 𝑞 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=% \left(b{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{n}{\mathrm{e}^{\mathrm{i}n% \theta}}\,\qHyperrphis{2}{1}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}}}{b^{-1}% q^{-n+1}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{b^{-1}q{\mathrm{e}^{-\mathrm{i}% \theta}}}}}} {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\qPochhammer{b\expe^{-\iunit\theta}}{q}{n}\expe^{\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},a\expe^{\iunit\theta}}{b^{-1}q^{-n+1}\expe^{\iunit\theta}}{q}{b^{-1}q\expe^{-\iunit\theta}} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ Q m ⁡ ( x ; a , b | q ) ⁢ Q n ⁡ ( x ; a , b | q ) ⁢ 𝑑 x = δ m , n ( q n + 1 , a ⁢ b ⁢ q n ; q ) ∞ 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 Al-Salam-Chihara-polynomial-Q 𝑚 𝑥 𝑎 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 differential-d 𝑥 Kronecker-delta 𝑚 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}Q_{m}\!\left(x;a,b\,|\,q\right)Q_{n}\!\left(x;a,b\,|\,q% \right)\,dx=\frac{\,\delta_{m,n}}{\left(q^{n+1},abq^{n};q\right)_{\infty}}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AlSalamChihara{m}@{x}{a}{b}{q}\AlSalamChihara{n}@{x}{a}{b}{q}\,dx =\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1},abq^n}{q}{\infty}} }

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

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

Substitution(s): w k = ( a - 2 ; q ) ∞ ( q , a ⁢ b , a - 1 ⁢ b ; q ) ∞ ⁢ ( 1 - a 2 ⁢ q 2 ⁢ k ) ⁢ ( a 2 , a ⁢ b ; q ) k ( 1 - a 2 ) ⁢ ( q , a ⁢ b - 1 ⁢ q ; q ) k ⁢ q - k 2 ⁢ ( 1 a 3 ⁢ b ) k subscript 𝑤 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑞 q-Pochhammer-symbol 𝑞 𝑎 𝑏 superscript 𝑎 1 𝑏 𝑞 1 superscript 𝑎 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑎 𝑏 𝑞 𝑘 1 superscript 𝑎 2 q-Pochhammer-symbol 𝑞 𝑎 superscript 𝑏 1 𝑞 𝑞 𝑘 superscript 𝑞 superscript 𝑘 2 superscript 1 superscript 𝑎 3 𝑏 𝑘 {\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{% \infty}}{\left(q,ab,a^{-1}b;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})\left(a^{2}% ,ab;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q;q\right)_{k}}q^{-k^{2}}\left(\frac% {1}{a^{3}b}\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 , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 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 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\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,b)% }}}} &
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 ⁢ Q n ⁡ ( x ) = Q n + 1 ⁡ ( x ) + ( a + b ) ⁢ q n ⁢ Q n ⁡ ( x ) + ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n - 1 ) ⁢ Q n - 1 ⁡ ( x ) 2 𝑥 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 1 𝑥 𝑎 𝑏 𝑞 𝑎 𝑏 superscript 𝑞 𝑛 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 Al-Salam-Chihara-polynomial-Q 𝑛 1 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xQ_{n}\!\left(x\right)=Q_{n+1}\!% \left(x\right)+(a+b)q^{n}Q_{n}\!\left(x\right)+(1-q^{n})(1-abq^{n-1})Q_{n-1}\!% \left(x\right)}}} {\displaystyle 2x\AlSalamChihara{n}@@{x}{a}{b}{q}=\AlSalamChihara{n+1}@@{x}{a}{b}{q}+(a+b)q^n\AlSalamChihara{n}@@{x}{a}{b}{q}+(1-q^n)(1-abq^{n-1})\AlSalamChihara{n-1}@@{x}{a}{b}{q} }
Q n ⁡ ( x ) := Q n ⁡ ( x ; a , b | q ) assign Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x\right):=Q_{n}\!\left% (x;a,b\,|\,q\right)}}} {\displaystyle \AlSalamChihara{n}@@{x}{a}{b}{q}:=\AlSalamChihara{n}@{x}{a}{b}{q} }

Monic recurrence relation

x ⁢ Q ^ n ⁡ ( x ; a , b | q ) = Q ^ n + 1 ⁡ ( x ; a , b | q ) + 1 2 ⁢ ( a + b ) ⁢ q n ⁢ Q ^ n ⁡ ( x ; a , b | q ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n - 1 ) ⁢ Q ^ n - 1 ⁡ ( x ; a , b | q ) 𝑥 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 1 2 𝑎 𝑏 superscript 𝑞 𝑛 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 1 4 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x;a,b\,|% \,q\right)={\widehat{Q}}_{n+1}\!\left(x;a,b\,|\,q\right)+\frac{1}{2}(a+b)q^{n}% {\widehat{Q}}_{n}\!\left(x;a,b\,|\,q\right)+\frac{1}{4}(1-q^{n})(1-abq^{n-1}){% \widehat{Q}}_{n-1}\!\left(x;a,b\,|\,q\right)}}} {\displaystyle x\monicAlSalamChihara{n}@@{x}{a}{b}{q}=\monicAlSalamChihara{n+1}@@{x}{a}{b}{q}+\frac{1}{2}(a+b)q^n\monicAlSalamChihara{n}@@{x}{a}{b}{q}+\frac{1}{4}(1-q^n)(1-abq^{n-1})\monicAlSalamChihara{n-1}@@{x}{a}{b}{q} }
Q n ⁡ ( x ; a , b | q ) = 2 n ⁢ Q ^ n ⁡ ( x ; a , b | q ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 2 𝑛 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=2^{% n}{\widehat{Q}}_{n}\!\left(x;a,b\,|\,q\right)}}} {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=2^n\monicAlSalamChihara{n}@@{x}{a}{b}{q} }

q-Difference equation

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

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

y ⁢ ( x ) = Q n ⁡ ( x ; a , b | q ) 𝑦 𝑥 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=Q_{n}\!\left(x;a,b\,|\,q\right% )}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 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 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\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,b)% }}}} &
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 ⁢ b ; q ) n a n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ z , a ⁢ z - 1 ⁢ a ⁢ b , 0 ⁢ q ⁢ q assign \LegendrePoly 𝑛 @ 𝑧 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 𝑧 𝑎 superscript 𝑧 1 𝑎 𝑏 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{z}:=\frac{\left(ab% ;q\right)_{n}}{a^{n}}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,0}{q}{q}}}} {\displaystyle \LegendrePoly{n}@{z}:=\frac{\qPochhammer{ab}{q}{n}}{a^n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,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 - b ⁢ z ) ( 1 - z 2 ) ⁢ ( 1 - q ⁢ z 2 ) 𝐴 𝑧 1 𝑎 𝑧 1 𝑏 𝑧 1 superscript 𝑧 2 1 𝑞 superscript 𝑧 2 {\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)}{(1-z^{2})(% 1-qz^{2})}}}}


Forward shift operator

δ q ⁢ Q n ⁡ ( x ; a , b | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - q n ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ Q n - 1 ⁢ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 | q ) subscript 𝛿 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 imaginary-unit 𝜃 imaginary-unit 𝜃 subscript 𝑄 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}Q_{n}\!\left(x;a,b\,|\,q% \right){}=-q^{-\frac{1}{2}n}(1-q^{n})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm% {e}^{-\mathrm{i}\theta}})Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q)}}} {\displaystyle \delta_q\AlSalamChihara{n}@{x}{a}{b}{q} {}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q) }

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


D q ⁢ Q n ⁡ ( x ; a , b | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) ⁢ 1 - q n 1 - q ⁢ Q n - 1 ⁢ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 | q ) subscript 𝐷 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 subscript 𝑄 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}Q_{n}\!\left(x;a,b\,|\,q\right% )=2q^{-\frac{1}{2}(n-1)}\frac{1-q^{n}}{1-q}Q_{n-1}(x;aq^{\frac{1}{2}},bq^{% \frac{1}{2}}|q)}}} {\displaystyle D_q\AlSalamChihara{n}@{x}{a}{b}{q}=2q^{-\frac{1}{2}(n-1)} \frac{1-q^n}{1-q}Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q) }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; a , b | q ) ⁢ Q n ⁡ ( x ; a , b | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 | q ) ⁢ Q n + 1 ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 1 2 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 subscript 𝑄 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b|q)% Q_{n}\!\left(x;a,b\,|\,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}},bq^{-\frac{1}{2}}|q){}Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q)}}} {\displaystyle \delta_q\left[{\tilde w}(x;a,b|q)\AlSalamChihara{n}@{x}{a}{b}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q) {} Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q) }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 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 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\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,b)% }}}} &
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 , b | q ) ⁢ Q n ⁡ ( x ; a , b | q ) ] = - 2 ⁢ q - 1 2 ⁢ n 1 - q ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 | q ) ⁢ Q n + 1 ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 2 superscript 𝑞 1 2 𝑛 1 𝑞 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 subscript 𝑄 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a,b|q)Q_{n% }\!\left(x;a,b\,|\,q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}% }(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q)Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-% \frac{1}{2}}|q)}}} {\displaystyle D_q\left[{\tilde w}(x;a,b|q)\AlSalamChihara{n}@{x}{a}{b}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q) Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q) }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 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 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\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,b)% }}}} &
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 , b | q ) ⁢ Q n ⁡ ( x ; a , b | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n , b ⁢ q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑛 conditional 𝑏 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q)Q_{n}\!\left(x;a% ,b\,|\,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;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n}|q)\right]% }}} {\displaystyle {\tilde w}(x;a,b|q)\AlSalamChihara{n}@{x}{a}{b}{q} {}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)} \left(D_q\right)^n\left[{\tilde w}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n}|q)\right] }

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

w ⁢ ( x ) := w ⁢ ( x ; a , b | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 conditional 𝑏 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 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 ℎ 𝑥 𝑏 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left(% {\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\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,b)% }}}} &
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 , b ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t , e - i ⁢ θ ⁢ t ; q ) ∞ = ∑ n = 0 ∞ Q n ⁡ ( x ; a , b | q ) ( q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑎 𝑡 𝑏 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 imaginary-unit 𝜃 𝑡 𝑞 superscript subscript 𝑛 0 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(at,bt;q\right)_{\infty}}% {\left({\mathrm{e}^{\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm{i}\theta}}t;q% \right)_{\infty}}=\sum_{n=0}^{\infty}\frac{Q_{n}\!\left(x;a,b\,|\,q\right)}{% \left(q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{at,bt}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\AlSalamChihara{n}@{x}{a}{b}{q}}{\qPochhammer{q}{q}{n}}t^n }

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


1 ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ a ⁢ b ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ Q n ⁡ ( x ; a , b | q ) ( a ⁢ b , q ; q ) n ⁢ t n 1 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑎 𝑏 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{% i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}% \theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab}{q}{{\mathrm{e}^{-\mathrm{i}% \theta}}t}=\sum_{n=0}^{\infty}\frac{Q_{n}\!\left(x;a,b\,|\,q\right)}{\left(ab,% q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\AlSalamChihara{n}@{x}{a}{b}{q}}{\qPochhammer{ab,q}{q}{n}}t^n }

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


( t ; q ) ∞ ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ ⁢ a ⁢ b ⁢ q ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ a n ⁢ q \binomial ⁢ n ⁢ 2 ( a ⁢ b , q ; q ) n ⁢ Q n ⁡ ( x ; a , b | q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 𝑎 𝑏 𝑞 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑞 𝑛 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot% \qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i% }\theta}}}{ab}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}a^{n}q^{\binomial{n}{2% }}}{\left(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^{n}}}} {\displaystyle \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab}{q}{t} {}=\sum_{n=0}^{\infty}\frac{(-1)^na^nq^{\binomial{n}{2}}}{\qPochhammer{ab,q}{q}{n}}\AlSalamChihara{n}@{x}{a}{b}{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 ⁢ 32 ⁢ @ ⁢ @ ⁢ γ , a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ a ⁢ b , γ ⁢ e i ⁢ θ ⁢ t ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( γ ; q ) n ( a ⁢ b , q ; q ) n ⁢ Q n ⁡ ( x ; a , b | q ) ⁢ t n q-Pochhammer-symbol 𝛾 imaginary-unit 𝜃 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 32 @ @ 𝛾 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑎 𝑏 𝛾 imaginary-unit 𝜃 𝑡 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑞 𝑛 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 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{3}{2}@@{\gamma,a{\mathrm{e}^{\mathrm{i}\theta% }},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab,\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(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^% {n}}}} {\displaystyle \frac{\qPochhammer{\gamma \expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab,\gamma \expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{ab,q}{q}{n}}\AlSalamChihara{n}@{x}{a}{b}{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 dual q-Hahn polynomial to Al-Salam-Chihara polynomial

p n ⁡ ( x ; a , b , 0 | q ) = Q n ⁡ ( x ; a , b | q ) continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 0 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,0|q\right)=Q_{n}% \!\left(x;a,b\,|\,q\right)}}} {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{0}{q}=\AlSalamChihara{n}@{x}{a}{b}{q} }

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} }

Al-Salam-Chihara polynomial to Continuous q-Laguerre polynomial

Q n ⁡ ( x ; q 1 2 ⁢ α + 1 4 , q 1 2 ⁢ α + 3 4 | q ) = ( q ; q ) n q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ P n ( α ) ⁡ ( x | q ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 3 4 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Laguerre-polynomial-P 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;q^{\frac{1}{2}\alpha% +\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}}\,|\,q\right)=\frac{\left(q;q% \right)_{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha)}_{n}\!\left(x|q% \right)}}} {\displaystyle \AlSalamChihara{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{q} =\frac{\qPochhammer{q}{q}{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqLaguerre{\alpha}{n}@{x}{q} }

Al-Salam-Chihara polynomial to Meixner-Pollaczek polynomial

lim q → 1 ⁡ Q n ⁡ ( cos ⁡ ( ln ⁡ q x + ϕ ) ; q λ ⁢ e i ⁢ ϕ , q λ ⁢ e - i ⁢ ϕ | q ) ( q ; q ) n = P n ( λ ) ⁡ ( x ; ϕ ) subscript → 𝑞 1 Al-Salam-Chihara-polynomial-Q 𝑛 superscript 𝑞 𝑥 italic-ϕ superscript 𝑞 𝜆 imaginary-unit italic-ϕ superscript 𝑞 𝜆 imaginary-unit italic-ϕ 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{Q_{n}\!% \left(\cos\left(\ln q^{x}+\phi\right);q^{\lambda}{\mathrm{e}^{\mathrm{i}\phi}}% ,q^{\lambda}{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)}{\left(q;q\right)_{n}}% =P^{(\lambda)}_{n}\!\left(x;\phi\right)}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamChihara{n}@{\cos@{\ln@@{q^x}+\phi}}{ q^{\lambda}\expe^{\iunit\phi}}{q^{\lambda}\expe^{-\iunit\phi}}{q}}{\qPochhammer{q}{q}{n}}=\MeixnerPollaczek{\lambda}{n}@{x}{\phi} }

Koornwinder Addendum: Al-Salam-Chihara

Re: (14.8.1)

Q n ⁡ ( 1 2 ⁢ ( a ⁢ q - x + a - 1 ⁢ q x ) ; a , b ; q - 1 ) = ( - 1 ) n ⁢ b n ⁢ q - 1 2 ⁢ n ⁢ ( n - 1 ) ⁢ ( ( a ⁢ b ) - 1 ; q ) n ⁢ \qHyperrphis ⁢ 31 ⁢ @ ⁢ @ ⁢ q - n , q - x , a - 2 ⁢ q x ⁢ ( a ⁢ b ) - 1 ⁢ q ⁢ q n ⁢ a ⁢ b - 1 Al-Salam-Chihara-polynomial-Q 𝑛 1 2 𝑎 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑥 𝑎 𝑏 superscript 𝑞 1 superscript 1 𝑛 superscript 𝑏 𝑛 superscript 𝑞 1 2 𝑛 𝑛 1 q-Pochhammer-symbol superscript 𝑎 𝑏 1 𝑞 𝑛 \qHyperrphis 31 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑥 superscript 𝑎 2 superscript 𝑞 𝑥 superscript 𝑎 𝑏 1 𝑞 superscript 𝑞 𝑛 𝑎 superscript 𝑏 1 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^% {-1}q^{x});a,b\,;\,q^{-1}\right)=(-1)^{n}b^{n}q^{-\frac{1}{2}n(n-1)}\left((ab)% ^{-1};q\right)_{n}\qHyperrphis{3}{1}@@{q^{-n},q^{-x},a^{-2}q^{x}}{(ab)^{-1}}{q% }{q^{n}ab^{-1}}}}} {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}}= (-1)^n b^n q^{-\frac12 n(n-1)}\qPochhammer{(ab)^{-1}}{q}{n} \qHyperrphis{3}{1}@@{q^{-n},q^{-x},a^{-2}q^x}{(ab)^{-1}}{q}{q^nab^{-1}} }
Q n ⁡ ( 1 2 ⁢ ( a ⁢ q - x + a - 1 ⁢ q x ) ; a , b ; q - 1 ) = ( - a ⁢ b - 1 ) x ⁢ q - 1 2 ⁢ x ⁢ ( x + 1 ) ⁢ ( q ⁢ b ⁢ a - 1 ; q ) x ( a - 1 ⁢ b - 1 ; q ) x ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , a - 2 ⁢ q x ⁢ q ⁢ b ⁢ a - 1 ⁢ q ⁢ q n + 1 Al-Salam-Chihara-polynomial-Q 𝑛 1 2 𝑎 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑥 𝑎 𝑏 superscript 𝑞 1 superscript 𝑎 superscript 𝑏 1 𝑥 superscript 𝑞 1 2 𝑥 𝑥 1 q-Pochhammer-symbol 𝑞 𝑏 superscript 𝑎 1 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑏 1 𝑞 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑥 superscript 𝑎 2 superscript 𝑞 𝑥 𝑞 𝑏 superscript 𝑎 1 𝑞 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^% {-1}q^{x});a,b\,;\,q^{-1}\right)=(-ab^{-1})^{x}q^{-\frac{1}{2}x(x+1)}\frac{% \left(qba^{-1};q\right)_{x}}{\left(a^{-1}b^{-1};q\right)_{x}}\qHyperrphis{2}{1% }@@{q^{-x},a^{-2}q^{x}}{qba^{-1}}{q}{q^{n+1}}}}} {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =(-ab^{-1})^x q^{-\frac12 x(x+1)} \frac{\qPochhammer{qba^{-1}}{q}{x}}{\qPochhammer{a^{-1}b^{-1}}{q}{x}} \qHyperrphis{2}{1}@@{q^{-x},a^{-2}q^x}{qba^{-1}}{q}{q^{n+1}} }
Q n ⁡ ( 1 2 ⁢ ( a ⁢ q - x + a - 1 ⁢ q x ) ; a , b ; q - 1 ) = ( - a ⁢ b - 1 ) x ⁢ q - 1 2 ⁢ x ⁢ ( x + 1 ) ⁢ ( q ⁢ b ⁢ a - 1 ; q ) x ( a - 1 ⁢ b - 1 ; q ) x ⁢ p x ⁡ ( q n ; b ⁢ a - 1 , ( q ⁢ a ⁢ b ) - 1 ; q ) Al-Salam-Chihara-polynomial-Q 𝑛 1 2 𝑎 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑥 𝑎 𝑏 superscript 𝑞 1 superscript 𝑎 superscript 𝑏 1 𝑥 superscript 𝑞 1 2 𝑥 𝑥 1 q-Pochhammer-symbol 𝑞 𝑏 superscript 𝑎 1 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑏 1 𝑞 𝑥 little-q-Jacobi-polynomial-p 𝑥 superscript 𝑞 𝑛 𝑏 superscript 𝑎 1 superscript 𝑞 𝑎 𝑏 1 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^% {-1}q^{x});a,b\,;\,q^{-1}\right)=(-ab^{-1})^{x}q^{-\frac{1}{2}x(x+1)}\frac{% \left(qba^{-1};q\right)_{x}}{\left(a^{-1}b^{-1};q\right)_{x}}p_{x}\!\left(q^{n% };ba^{-1},(qab)^{-1};q\right)}}} {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =(-ab^{-1})^x q^{-\frac12 x(x+1)} \frac{\qPochhammer{qba^{-1}}{q}{x}}{\qPochhammer{a^{-1}b^{-1}}{q}{x}} \littleqJacobi{x}@{q^n}{ba^{-1}}{(qab)^{-1}}{q} }

Orthogonality

∑ x = 0 ∞ ( 1 - q 2 ⁢ x ⁢ a - 2 ) ⁢ ( a - 2 , ( a ⁢ b ) - 1 ; q ) x ( 1 - a - 2 ) ⁢ ( q , b ⁢ q ⁢ a - 1 ; q ) x ( b a - 1 ) x q x 2 Q m ⁡ ( 1 2 ⁢ ( a ⁢ q - x + a - 1 ⁢ q x ) ; a , b ; q - 1 ) Q n ⁡ ( 1 2 ⁢ ( a ⁢ q - x + a - 1 ⁢ q x ) ; a , b ; q - 1 ) = ( q ⁢ a - 2 ; q ) ∞ ( b ⁢ a - 1 ⁢ q ; q ) ∞ ( q , ( a ⁢ b ) - 1 ; q ) n ( a b ) n q - n 2 δ m , n ( a b > 1 , q b < a ) fragments superscript subscript 𝑥 0 1 superscript 𝑞 2 𝑥 superscript 𝑎 2 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑎 𝑏 1 𝑞 𝑥 1 superscript 𝑎 2 q-Pochhammer-symbol 𝑞 𝑏 𝑞 superscript 𝑎 1 𝑞 𝑥 superscript fragments ( b superscript 𝑎 1 ) 𝑥 superscript 𝑞 superscript 𝑥 2 Al-Salam-Chihara-polynomial-Q 𝑚 1 2 𝑎 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑥 𝑎 𝑏 superscript 𝑞 1 Al-Salam-Chihara-polynomial-Q 𝑛 1 2 𝑎 superscript 𝑞 𝑥 superscript 𝑎 1 superscript 𝑞 𝑥 𝑎 𝑏 superscript 𝑞 1 q-Pochhammer-symbol 𝑞 superscript 𝑎 2 𝑞 q-Pochhammer-symbol 𝑏 superscript 𝑎 1 𝑞 𝑞 q-Pochhammer-symbol 𝑞 superscript 𝑎 𝑏 1 𝑞 𝑛 superscript fragments ( a b ) 𝑛 superscript 𝑞 superscript 𝑛 2 Kronecker-delta 𝑚 𝑛 fragments ( a b 1 , q b a ) {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{(1-q^{2x}a^% {-2})\left(a^{-2},(ab)^{-1};q\right)_{x}}{(1-a^{-2})\left(q,bqa^{-1};q\right)_% {x}}(ba^{-1})^{x}q^{x^{2}}Q_{m}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;% \,q^{-1}\right)Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}% \right)=\frac{\left(qa^{-2};q\right)_{\infty}}{\left(ba^{-1}q;q\right)_{\infty% }}\left(q,(ab)^{-1};q\right)_{n}(ab)^{n}q^{-n^{2}}\delta_{m,n}(ab>1,\;qb<a)}}} {\displaystyle \sum_{x=0}^\infty \frac{(1-q^{2x}a^{-2}) \qPochhammer{a^{-2},(ab)^{-1}}{q}{x}} {(1-a^{-2}) \qPochhammer{q,bqa^{-1}}{q}{x}} (ba^{-1})^xq^{x^2} \qinvAlSalamChihara{m}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}}\qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =\frac{\qPochhammer{qa^{-2}}{q}{\infty}}{\qPochhammer{ba^{-1}q}{q}{\infty}} \qPochhammer{q,(ab)^{-1}}{q}{n} (ab)^nq^{-n^2} \Kronecker{m}{n} (ab>1,\;qb

Normalized recurrence relation

x ⁢ Q ^ n ⁡ ( x ) = Q ^ n + 1 ⁡ ( x ) + 1 2 ⁢ ( a + b ) ⁢ q - n ⁢ Q ^ n ⁡ ( x ) + 1 4 ⁢ ( q - n - 1 ) ⁢ ( a ⁢ b ⁢ q - n + 1 - 1 ) ⁢ Q ^ n - 1 ⁡ ( x ) 𝑥 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 superscript 𝑞 1 1 2 𝑎 𝑏 superscript 𝑞 𝑛 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 1 4 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 Al-Salam-Chihara-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 superscript 𝑞 1 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)=% {\widehat{Q}}_{n+1}\!\left(x\right)+\frac{1}{2}(a+b)q^{-n}{\widehat{Q}}_{n}\!% \left(x\right)+\tfrac{1}{4}(q^{-n}-1)(abq^{-n+1}-1){\widehat{Q}}_{n-1}\!\left(% x\right)}}} {\displaystyle x\monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}}=\monicqinvAlSalamChihara{n+1}@@{x}{a}{b }{q^{-1}}+\frac12(a+b)q^{-n} \monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}}+ \tfrac14(q^{-n}-1)(abq^{-n+1}-1)\monicqinvAlSalamChihara{n-1}@@{x}{a}{b }{q^{-1}} }
Q n ⁡ ( x ; a , b ; q - 1 ) = 2 n ⁢ Q ^ n ⁡ ( x ) Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 superscript 2 𝑛 Al-Salam-Chihara-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 superscript 𝑞 1 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,;\,q^{-1}\right% )=2^{n}{\widehat{Q}}_{n}\!\left(x\right)}}} {\displaystyle \qinvAlSalamChihara{n}@{x}{a}{b }{q^{-1}}=2^n \monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}} }

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