Al-Salam-Carlitz I

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Al-Salam-Carlitz I

Basic hypergeometric representation

U n ( a ) ⁡ ( x ; q ) = ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , x - 1 ⁢ 0 ⁢ q ⁢ q ⁢ x a q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑥 1 0 𝑞 𝑞 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)=(-a)^% {n}q^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}% }}}} {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}=(-a)^nq^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}} }

Orthogonality relation(s)

∫ a 1 ( q ⁢ x , a - 1 ⁢ q ⁢ x ; q ) ∞ ⁢ U m ( a ) ⁡ ( x ; q ) ⁢ U n ( a ) ⁡ ( x ; q ) ⁢ d q ⁢ x = ( - a ) n ⁢ ( 1 - q ) ⁢ ( q ; q ) n ⁢ ( q , a , a - 1 ⁢ q ; q ) ∞ ⁢ q \binomial ⁢ n ⁢ 2 ⁢ δ m , n superscript subscript 𝑎 1 q-Pochhammer-symbol 𝑞 𝑥 superscript 𝑎 1 𝑞 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑚 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 subscript 𝑑 𝑞 𝑥 superscript 𝑎 𝑛 1 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑎 superscript 𝑎 1 𝑞 𝑞 superscript 𝑞 \binomial 𝑛 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{1}\left(qx,a^{-1}qx;q% \right)_{\infty}U^{(a)}_{m}\!\left(x;q\right)U^{(a)}_{n}\!\left(x;q\right)\,d_% {q}x{}=(-a)^{n}(1-q)\left(q;q\right)_{n}\left(q,a,a^{-1}q;q\right)_{\infty}q^{% \binomial{n}{2}}\,\delta_{m,n}}}} {\displaystyle \int_a^1\qPochhammer{qx,a^{-1}qx}{q}{\infty}\AlSalamCarlitzI{a}{m}@{x}{q}\AlSalamCarlitzI{a}{n}@{x}{q}\,d_qx {}=(-a)^n(1-q)\qPochhammer{q}{q}{n}\qPochhammer{q,a,a^{-1}q}{q}{\infty}q^{\binomial{n}{2}}\,\Kronecker{m}{n} }

Constraint(s): a < 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a<0}}}


Recurrence relation

x ⁢ U n ( a ) ⁡ ( x ; q ) = U n + 1 ( a ) ⁡ ( x ; q ) + ( a + 1 ) ⁢ q n ⁢ U n ( a ) ⁡ ( x ; q ) - a ⁢ q n - 1 ⁢ ( 1 - q n ) ⁢ U n - 1 ( a ) ⁡ ( x ; q ) 𝑥 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 𝑎 1 superscript 𝑞 𝑛 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 𝑎 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle xU^{(a)}_{n}\!\left(x;q\right)=U^{(% a)}_{n+1}\!\left(x;q\right)+(a+1)q^{n}U^{(a)}_{n}\!\left(x;q\right)-aq^{n-1}(1% -q^{n})U^{(a)}_{n-1}\!\left(x;q\right)}}} {\displaystyle x\AlSalamCarlitzI{a}{n}@{x}{q}=\AlSalamCarlitzI{a}{n+1}@{x}{q}+(a+1)q^n\AlSalamCarlitzI{a}{n}@{x}{q} -aq^{n-1}(1-q^n)\AlSalamCarlitzI{a}{n-1}@{x}{q} }

Monic recurrence relation

x ⁢ U ^ n ( a ) ⁡ ( x ) = U ^ n + 1 ( a ) ⁡ ( x ) + ( a + 1 ) ⁢ q n ⁢ U ^ n ( a ) ⁡ ( x ) - a ⁢ q n - 1 ⁢ ( 1 - q n ) ⁢ U ^ n - 1 ( a ) ⁡ ( x ) 𝑥 q-Al-Salam-Carlitz-I-polynomial-monic-p 𝑎 𝑛 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-monic-p 𝑎 𝑛 1 𝑥 𝑞 𝑎 1 superscript 𝑞 𝑛 q-Al-Salam-Carlitz-I-polynomial-monic-p 𝑎 𝑛 𝑥 𝑞 𝑎 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 q-Al-Salam-Carlitz-I-polynomial-monic-p 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{U}}^{(a)}_{n}\!\left(x% \right)={\widehat{U}}^{(a)}_{n+1}\!\left(x\right)+(a+1)q^{n}{\widehat{U}}^{(a)% }_{n}\!\left(x\right)-aq^{n-1}(1-q^{n}){\widehat{U}}^{(a)}_{n-1}\!\left(x% \right)}}} {\displaystyle x\monicAlSalamCarlitzI{a}{n}@@{x}{q}=\monicAlSalamCarlitzI{a}{n+1}@@{x}{q}+(a+1)q^n\monicAlSalamCarlitzI{a}{n}@@{x}{q}-aq^{n-1}(1-q^n)\monicAlSalamCarlitzI{a}{n-1}@@{x}{q} }
U n ( a ) ⁡ ( x ; q ) = U ^ n ( a ) ⁡ ( x ) q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-monic-p 𝑎 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)={% \widehat{U}}^{(a)}_{n}\!\left(x\right)}}} {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}=\monicAlSalamCarlitzI{a}{n}@@{x}{q} }

q-Difference equation

( 1 - q n ) ⁢ x 2 ⁢ y ⁢ ( x ) = a ⁢ q n - 1 ⁢ y ⁢ ( q ⁢ x ) - [ a ⁢ q n - 1 + q n ⁢ ( 1 - x ) ⁢ ( a - x ) ] ⁢ y ⁢ ( x ) + q n ⁢ ( 1 - x ) ⁢ ( a - x ) ⁢ y ⁢ ( q - 1 ⁢ x ) 1 superscript 𝑞 𝑛 superscript 𝑥 2 𝑦 𝑥 𝑎 superscript 𝑞 𝑛 1 𝑦 𝑞 𝑥 delimited-[] 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑥 𝑎 𝑥 𝑦 𝑥 superscript 𝑞 𝑛 1 𝑥 𝑎 𝑥 𝑦 superscript 𝑞 1 𝑥 {\displaystyle{\displaystyle{\displaystyle(1-q^{n})x^{2}y(x)=aq^{n-1}y(qx)-% \left[aq^{n-1}+q^{n}(1-x)(a-x)\right]y(x){}+q^{n}(1-x)(a-x)y(q^{-1}x)}}} {\displaystyle (1-q^n)x^2y(x)=aq^{n-1}y(qx)-\left[aq^{n-1}+q^n(1-x)(a-x)\right]y(x) {}+q^n(1-x)(a-x)y(q^{-1}x) }

Substitution(s): y ⁢ ( x ) = U n ( a ) ⁡ ( x ; q ) 𝑦 𝑥 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=U^{(a)}_{n}\!\left(x;q\right)}}}


Forward shift operator

U n ( a ) ⁡ ( x ; q ) - U n ( a ) ⁡ ( q ⁢ x ; q ) = ( 1 - q n ) ⁢ x ⁢ U n - 1 ( a ) ⁡ ( x ; q ) q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑞 𝑥 𝑞 1 superscript 𝑞 𝑛 𝑥 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)-U^{(a% )}_{n}\!\left(qx;q\right)=(1-q^{n})xU^{(a)}_{n-1}\!\left(x;q\right)}}} {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q}-\AlSalamCarlitzI{a}{n}@{qx}{q}=(1-q^n)x\AlSalamCarlitzI{a}{n-1}@{x}{q} }
𝒟 q ⁢ U n ( a ) ⁡ ( x ; q ) = 1 - q n 1 - q ⁢ U n - 1 ( a ) ⁡ ( x ; q ) q-derivative 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 1 superscript 𝑞 𝑛 1 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}U^{(a)}_{n}\!\left(x;% q\right)=\frac{1-q^{n}}{1-q}U^{(a)}_{n-1}\!\left(x;q\right)}}} {\displaystyle \qderiv{q}\AlSalamCarlitzI{a}{n}@{x}{q}=\frac{1-q^n}{1-q}\AlSalamCarlitzI{a}{n-1}@{x}{q} }

Backward shift operator

a ⁢ U n ( a ) ⁡ ( x ; q ) - ( 1 - x ) ⁢ ( a - x ) ⁢ U n ( a ) ⁡ ( q - 1 ⁢ x ; q ) = - q - n ⁢ x ⁢ U n + 1 ( a ) ⁡ ( x ; q ) 𝑎 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 1 𝑥 𝑎 𝑥 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 superscript 𝑞 1 𝑥 𝑞 superscript 𝑞 𝑛 𝑥 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle aU^{(a)}_{n}\!\left(x;q\right)-(1-x% )(a-x)U^{(a)}_{n}\!\left(q^{-1}x;q\right)=-q^{-n}xU^{(a)}_{n+1}\!\left(x;q% \right)}}} {\displaystyle a\AlSalamCarlitzI{a}{n}@{x}{q}-(1-x)(a-x)\AlSalamCarlitzI{a}{n}@{q^{-1}x}{q}=-q^{-n}x\AlSalamCarlitzI{a}{n+1}@{x}{q} }
𝒟 q - 1 ⁢ [ w ⁢ ( x ; a ; q ) ⁢ U n ( a ) ⁡ ( x ; q ) ] = q - n + 1 a ⁢ ( 1 - q ) ⁢ w ⁢ ( x ; a ; q ) ⁢ U n + 1 ( a ) ⁡ ( x ; q ) subscript 𝒟 superscript 𝑞 1 delimited-[] 𝑤 𝑥 𝑎 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 superscript 𝑞 𝑛 1 𝑎 1 𝑞 𝑤 𝑥 𝑎 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q^{-1}}\left[w(x;a;q)U^% {(a)}_{n}\!\left(x;q\right)\right]=\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U^{(a)}_{n+1% }\!\left(x;q\right)}}} {\displaystyle \mathcal{D}_{q^{-1}}\left[w(x;a;q)\AlSalamCarlitzI{a}{n}@{x}{q}\right]= \frac{q^{-n+1}}{a(1-q)}w(x;a;q)\AlSalamCarlitzI{a}{n+1}@{x}{q} }

Substitution(s): w ⁢ ( x ; a ; q ) = ( q ⁢ x , a - 1 ⁢ q ⁢ x ; q ) ∞ 𝑤 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑥 superscript 𝑎 1 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\left(qx,a^{-1}qx;q\right)% _{\infty}}}}


Rodrigues-type formula

w ⁢ ( x ; a ; q ) ⁢ U n ( a ) ⁡ ( x ; q ) = a n ⁢ q 1 2 ⁢ n ⁢ ( n - 3 ) ⁢ ( 1 - q ) n ⁢ ( 𝒟 q - 1 ) n ⁢ [ w ⁢ ( x ; a ; q ) ] 𝑤 𝑥 𝑎 𝑞 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 superscript 𝑎 𝑛 superscript 𝑞 1 2 𝑛 𝑛 3 superscript 1 𝑞 𝑛 superscript subscript 𝒟 superscript 𝑞 1 𝑛 delimited-[] 𝑤 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;a;q)U^{(a)}_{n}\!\left(x;q% \right)=a^{n}q^{\frac{1}{2}n(n-3)}(1-q)^{n}\left(\mathcal{D}_{q^{-1}}\right)^{% n}\left[w(x;a;q)\right]}}} {\displaystyle w(x;a;q)\AlSalamCarlitzI{a}{n}@{x}{q}=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n \left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;a;q)\right] }

Substitution(s): w ⁢ ( x ; a ; q ) = ( q ⁢ x , a - 1 ⁢ q ⁢ x ; q ) ∞ 𝑤 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑥 superscript 𝑎 1 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\left(qx,a^{-1}qx;q\right)% _{\infty}}}}


Generating function

( t , a ⁢ t ; q ) ∞ ( x ⁢ t ; q ) ∞ = ∑ n = 0 ∞ U n ( a ) ⁡ ( x ; q ) ( q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑡 𝑎 𝑡 𝑞 q-Pochhammer-symbol 𝑥 𝑡 𝑞 superscript subscript 𝑛 0 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(t,at;q\right)_{\infty}}{% \left(xt;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{U^{(a)}_{n}\!\left(x;q% \right)}{\left(q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{t,at}{q}{\infty}}{\qPochhammer{xt}{q}{\infty}}= \sum_{n=0}^{\infty}\frac{\AlSalamCarlitzI{a}{n}@{x}{q}}{\qPochhammer{q}{q}{n}}t^n }

Limit relations

Big q-Laguerre polynomial to Al-Salam-Carlitz I polynomial

lim a → 0 ⁡ P n ⁡ ( a ⁢ q ⁢ x ; a , a ⁢ b ; q ) a n = q n ⁢ U n ( b ) ⁡ ( x ; q ) subscript → 𝑎 0 big-q-Laguerre-polynomial-P 𝑛 𝑎 𝑞 𝑥 𝑎 𝑎 𝑏 𝑞 superscript 𝑎 𝑛 superscript 𝑞 𝑛 q-Al-Salam-Carlitz-I-polynomial-U 𝑏 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}\frac{P_{n}\!% \left(aqx;a,ab;q\right)}{a^{n}}=q^{n}U^{(b)}_{n}\!\left(x;q\right)}}} {\displaystyle \lim_{a\rightarrow 0}\frac{\bigqLaguerre{n}@{aqx}{a}{ab}{q}}{a^n}=q^n\AlSalamCarlitzI{b}{n}@{x}{q} }

Dual q-Krawtchouk polynomial to Al-Salam-Carlitz I polynomial

lim N → ∞ ⁡ K n ⁡ ( λ ⁢ ( x ) ; a - 1 , N | q ) = ( - 1 a ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ U n ( a ) ⁡ ( q x ; q ) subscript → 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 superscript 𝑎 1 𝑁 𝑞 superscript 1 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 superscript 𝑞 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!% \left(\lambda(x);a^{-1},N|q\right)=\left(-\frac{1}{a}\right)^{n}q^{-\binomial{% n}{2}}U^{(a)}_{n}\!\left(q^{x};q\right)}}} {\displaystyle \lim_{N\rightarrow\infty}\dualqKrawtchouk{n}@{\lambda(x)}{a^{-1}}{N}{q}= \left(-\frac{1}{a}\right)^nq^{-\binomial{n}{2}}\AlSalamCarlitzI{a}{n}@{q^x}{q} }

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

U n ( - 1 ) ⁡ ( x ; q ) = h n ⁡ ( x ; q ) q-Al-Salam-Carlitz-I-polynomial-U 1 𝑛 𝑥 𝑞 discrete-q-Hermite-polynomial-h-I 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle U^{(-1)}_{n}\!\left(x;q\right)=h_{n% }\!\left(x;q\right)}}} {\displaystyle \AlSalamCarlitzI{-1}{n}@{x}{q}=\discrqHermiteI{n}@{x}{q} }

Al-Salam-Carlitz I polynomial to Charlier / Hermite polynomial

lim q → 1 ⁡ U n ( a ⁢ ( q - 1 ) ) ⁡ ( q x ; q ) ( 1 - q ) n = a n ⁢ C n ⁡ ( x ; a ) subscript → 𝑞 1 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑞 1 𝑛 superscript 𝑞 𝑥 𝑞 superscript 1 𝑞 𝑛 superscript 𝑎 𝑛 Charlier-polynomial-C 𝑛 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{U^{(a(q-1% ))}_{n}\!\left(q^{x};q\right)}{(1-q)^{n}}=a^{n}C_{n}\!\left(x;a\right)}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamCarlitzI{a(q-1)}{n}@{q^x}{q}}{(1-q)^n}=a^n\Charlier{n}@{x}{a} }
lim q → 1 ⁡ U n ( a ⁢ 1 - q 2 - 1 ) ⁡ ( x ⁢ 1 - q 2 ; q ) ( 1 - q 2 ) n 2 = H n ⁡ ( x - a ) 2 n subscript → 𝑞 1 q-Al-Salam-Carlitz-I-polynomial-U 𝑎 1 superscript 𝑞 2 1 𝑛 𝑥 1 superscript 𝑞 2 𝑞 superscript 1 superscript 𝑞 2 𝑛 2 Hermite-polynomial-H 𝑛 𝑥 𝑎 superscript 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{U^{(a% \sqrt{1-q^{2}}-1)}_{n}\!\left(x\sqrt{1-q^{2}};q\right)}{(1-q^{2})^{\frac{n}{2}% }}=\frac{H_{n}\left(x-a\right)}{2^{n}}}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamCarlitzI{a\sqrt{1-q^2}-1}{n}@{x\sqrt{1-q^2}}{q}} {(1-q^2)^{\frac{n}{2}}}=\frac{\Hermite{n}@{x-a}}{2^n} }

Remark

U n ( a ) ⁡ ( x ; q - 1 ) = V n ( a ) ⁡ ( x ; q ) q-Al-Salam-Carlitz-I-polynomial-U 𝑎 𝑛 𝑥 superscript 𝑞 1 q-Al-Salam-Carlitz-II-polynomial-V 𝑎 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q^{-1}\right)=% V^{(a)}_{n}\!\left(x;q\right)}}} {\displaystyle \AlSalamCarlitzI{a}{n}@{x}{q^{-1}}=\AlSalamCarlitzII{a}{n}@{x}{q} }

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