Little q-Jacobi

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Little q-Jacobi

Basic hypergeometric representation

p n ⁡ ( x ; a , b ; q ) = \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ b ⁢ q n + 1 ⁢ a ⁢ q ⁢ q ⁢ q ⁢ x little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑞 𝑞 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right)=% \qHyperrphis{2}{1}@@{q^{-n},abq^{n+1}}{aq}{q}{qx}}}} {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\qHyperrphis{2}{1}@@{q^{-n},abq^{n+1}}{aq}{q}{qx} }

Orthogonality relation(s)

∑ k = 0 ∞ ( b ⁢ q ; q ) k ( q ; q ) k ⁢ ( a ⁢ q ) k ⁢ p m ⁡ ( q k ; a , b ; q ) ⁢ p n ⁡ ( q k ; a , b ; q ) = ( a ⁢ b ⁢ q 2 ; q ) ∞ ( a ⁢ q ; q ) ∞ ⁢ ( 1 - a ⁢ b ⁢ q ) ⁢ ( a ⁢ q ) n ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , b ⁢ q ; q ) n ( a ⁢ q , a ⁢ b ⁢ q ; q ) n ⁢ δ m , n superscript subscript 𝑘 0 q-Pochhammer-symbol 𝑏 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 𝑎 𝑞 𝑘 little-q-Jacobi-polynomial-p 𝑚 superscript 𝑞 𝑘 𝑎 𝑏 𝑞 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑞 𝑘 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑏 superscript 𝑞 2 𝑞 q-Pochhammer-symbol 𝑎 𝑞 𝑞 1 𝑎 𝑏 𝑞 superscript 𝑎 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝑏 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑎 𝑏 𝑞 𝑞 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\left(bq;q% \right)_{k}}{\left(q;q\right)_{k}}(aq)^{k}p_{m}\!\left(q^{k};a,b;q\right)p_{n}% \!\left(q^{k};a,b;q\right){}=\frac{\left(abq^{2};q\right)_{\infty}}{\left(aq;q% \right)_{\infty}}\frac{(1-abq)(aq)^{n}}{(1-abq^{2n+1})}\frac{\left(q,bq;q% \right)_{n}}{\left(aq,abq;q\right)_{n}}\,\delta_{m,n}}}} {\displaystyle \sum_{k=0}^{\infty}\frac{\qPochhammer{bq}{q}{k}}{\qPochhammer{q}{q}{k}}(aq)^k\littleqJacobi{m}@{q^k}{a}{b}{q}\littleqJacobi{n}@{q^k}{a}{b}{q} {}=\frac{\qPochhammer{abq^2}{q}{\infty}}{\qPochhammer{aq}{q}{\infty}}\frac{(1-abq)(aq)^n}{(1-abq^{2n+1})} \frac{\qPochhammer{q,bq}{q}{n}}{\qPochhammer{aq,abq}{q}{n}}\,\Kronecker{m}{n} }

Recurrence relation

- x ⁢ p n ⁡ ( x ; a , b ; q ) = A n ⁢ p n + 1 ⁡ ( x ; a , b ; q ) - ( A n + C n ) ⁢ p n ⁡ ( x ; a , b ; q ) + C n ⁢ p n - 1 ⁡ ( x ; a , b ; q ) 𝑥 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 subscript 𝐴 𝑛 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 subscript 𝐶 𝑛 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x;a,b;q\right)=A_{n}p% _{n+1}\!\left(x;a,b;q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x;a,b;q% \right){}+C_{n}p_{n-1}\!\left(x;a,b;q\right)}}} {\displaystyle -x\littleqJacobi{n}@{x}{a}{b}{q}=A_n\littleqJacobi{n+1}@{x}{a}{b}{q}-\left(A_n+C_n\right)\littleqJacobi{n}@{x}{a}{b}{q} {}+C_n\littleqJacobi{n-1}@{x}{a}{b}{q} }

Substitution(s): C n = a ⁢ q n ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ q n ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{n}\frac{(1-q^{n})(1-bq^{n% })}{(1-abq^{2n})(1-abq^{2n+1})}}}} &
A n = q n ⁢ ( 1 - a ⁢ q n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1-aq^{n+1})(1-abq% ^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}}


Monic recurrence relation

x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ p ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 little-q-Jacobi-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 little-q-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Jacobi-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 little-q-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)=% {\widehat{p}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}){\widehat{p}}_{n}\!\left(x% \right)+A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}} {\displaystyle x\moniclittleqJacobi{n}@@{x}{a}{b}{q}=\moniclittleqJacobi{n+1}@@{x}{a}{b}{q}+(A_n+C_n)\moniclittleqJacobi{n}@@{x}{a}{b}{q}+A_{n-1}C_n\moniclittleqJacobi{n-1}@@{x}{a}{b}{q} }

Substitution(s): C n = a ⁢ q n ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ q n ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{n}\frac{(1-q^{n})(1-bq^{n% })}{(1-abq^{2n})(1-abq^{2n+1})}}}} &
A n = q n ⁢ ( 1 - a ⁢ q n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1-aq^{n+1})(1-abq% ^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}}


p n ⁡ ( x ; a , b ; q ) = ( - 1 ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ ( a ⁢ b ⁢ q n + 1 ; q ) n ( a ⁢ q ; q ) n ⁢ p ^ n ⁡ ( x ) little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 little-q-Jacobi-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right)=\frac{(% -1)^{n}q^{-\binomial{n}{2}}\left(abq^{n+1};q\right)_{n}}{\left(aq;q\right)_{n}% }{\widehat{p}}_{n}\!\left(x\right)}}} {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{abq^{n+1}}{q}{n}}{\qPochhammer{aq}{q}{n}}\moniclittleqJacobi{n}@@{x}{a}{b}{q} }

q-Difference equation

q - n ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ⁢ x ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( q ⁢ x ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( q - 1 ⁢ x ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑥 𝑦 𝑥 𝐵 𝑥 𝑦 𝑞 𝑥 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 superscript 𝑞 1 𝑥 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-abq^{n+1})xy(x){}% =B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x)}}} {\displaystyle q^{-n}(1-q^n)(1-abq^{n+1})xy(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }

Substitution(s): D ⁢ ( x ) = x - 1 𝐷 𝑥 𝑥 1 {\displaystyle{\displaystyle{\displaystyle D(x)=x-1}}} &

B ⁢ ( x ) = a ⁢ ( b ⁢ q ⁢ x - 1 ) 𝐵 𝑥 𝑎 𝑏 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=a(bqx-1)}}} &

y ⁢ ( x ) = p n ⁡ ( x ; a , b ; q ) 𝑦 𝑥 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b;q\right)}}}


Forward shift operator

p n ⁡ ( x ; a , b ; q ) - p n ⁡ ( q ⁢ x ; a , b ; q ) = - q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ( 1 - a ⁢ q ) ⁢ x ⁢ p n - 1 ⁡ ( x ; a ⁢ q , b ⁢ q ; q ) little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑞 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑎 𝑞 𝑥 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 𝑎 𝑞 𝑏 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right)-p_{n}\!% \left(qx;a,b;q\right){}=-\frac{q^{-n+1}(1-q^{n})(1-abq^{n+1})}{(1-aq)}xp_{n-1}% \!\left(x;aq,bq;q\right)}}} {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}-\littleqJacobi{n}@{qx}{a}{b}{q} {}=-\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})}{(1-aq)}x\littleqJacobi{n-1}@{x}{aq}{bq}{q} }
𝒟 q ⁢ p n ⁡ ( x ; a , b ; q ) = - q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ( 1 - q ) ⁢ ( 1 - a ⁢ q ) ⁢ p n - 1 ⁡ ( x ; a ⁢ q , b ⁢ q ; q ) q-derivative 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑞 1 𝑎 𝑞 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 𝑎 𝑞 𝑏 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}p_{n}\!\left(x;a,b;q% \right)=-\frac{q^{-n+1}(1-q^{n})(1-abq^{n+1})}{(1-q)(1-aq)}p_{n-1}\!\left(x;aq% ,bq;q\right)}}} {\displaystyle \qderiv{q}\littleqJacobi{n}@{x}{a}{b}{q}=-\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})} {(1-q)(1-aq)}\littleqJacobi{n-1}@{x}{aq}{bq}{q} }

Backward shift operator

a ⁢ ( b ⁢ x - 1 ) ⁢ p n ⁡ ( x ; a , b ; q ) - ( x - 1 ) ⁢ p n ⁡ ( q - 1 ⁢ x ; a , b ; q ) = ( 1 - a ) ⁢ p n + 1 ⁡ ( x ; a ⁢ q - 1 , b ⁢ q - 1 ; q ) 𝑎 𝑏 𝑥 1 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 𝑥 1 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑞 1 𝑥 𝑎 𝑏 𝑞 1 𝑎 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 𝑏 superscript 𝑞 1 𝑞 {\displaystyle{\displaystyle{\displaystyle a(bx-1)p_{n}\!\left(x;a,b;q\right)-% (x-1)p_{n}\!\left(q^{-1}x;a,b;q\right){}=(1-a)p_{n+1}\!\left(x;aq^{-1},bq^{-1}% ;q\right)}}} {\displaystyle a(bx-1)\littleqJacobi{n}@{x}{a}{b}{q}-(x-1)\littleqJacobi{n}@{q^{-1}x}{a}{b}{q} {}=(1-a)\littleqJacobi{n+1}@{x}{aq^{-1}}{bq^{-1}}{q} }
𝒟 q - 1 ⁢ [ w ⁢ ( x ; α , β | q ) ⁢ p n ⁡ ( x ; q α , q β ; q ) ] = 1 - q α q α - 1 ⁢ ( 1 - q ) ⁢ w ⁢ ( x ; α - 1 , β - 1 | q ) ⁢ p n + 1 ⁡ ( x ; q α - 1 , q β - 1 ; q ) subscript 𝒟 superscript 𝑞 1 delimited-[] 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 1 superscript 𝑞 𝛼 superscript 𝑞 𝛼 1 1 𝑞 𝑤 𝑥 𝛼 1 𝛽 conditional 1 𝑞 little-q-Jacobi-polynomial-p 𝑛 1 𝑥 superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q^{-1}}\left[w(x;\alpha% ,\beta|q)p_{n}\!\left(x;q^{\alpha},q^{\beta};q\right)\right]{}=\frac{1-q^{% \alpha}}{q^{\alpha-1}(1-q)}w(x;\alpha-1,\beta-1|q)p_{n+1}\!\left(x;q^{\alpha-1% },q^{\beta-1};q\right)}}} {\displaystyle \mathcal{D}_{q^{-1}}\left[w(x;\alpha,\beta|q)\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q}\right] {}=\frac{1-q^{\alpha}}{q^{\alpha-1}(1-q)}w(x;\alpha-1,\beta-1|q)\littleqJacobi{n+1}@{x}{q^{\alpha-1}}{q^{\beta-1}}{q} }

Substitution(s): w ⁢ ( x ; α , β | q ) = ( q ⁢ x ; q ) ∞ ( q β + 1 ⁢ x ; q ) ∞ ⁢ x α 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 q-Pochhammer-symbol 𝑞 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛽 1 𝑥 𝑞 superscript 𝑥 𝛼 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)=\frac{\left(qx;% q\right)_{\infty}}{\left(q^{\beta+1}x;q\right)_{\infty}}x^{\alpha}}}}


Rodrigues-type formula

w ⁢ ( x ; α , β | q ) ⁢ p n ⁡ ( x ; q α , q β ; q ) = q n ⁢ α + \binomial ⁢ n ⁢ 2 ⁢ ( 1 - q ) n ( q α + 1 ; q ) n ⁢ ( 𝒟 q - 1 ) n ⁢ [ w ⁢ ( x ; α + n , β + n | q ) ] 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 superscript 𝑞 𝑛 𝛼 \binomial 𝑛 2 superscript 1 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript subscript 𝒟 superscript 𝑞 1 𝑛 delimited-[] 𝑤 𝑥 𝛼 𝑛 𝛽 conditional 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)p_{n}\!\left(x;q% ^{\alpha},q^{\beta};q\right){}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^{n}}{% \left(q^{\alpha+1};q\right)_{n}}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[w(x% ;\alpha+n,\beta+n|q)\right]}}} {\displaystyle w(x;\alpha,\beta|q)\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q} {}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^n}{\qPochhammer{q^{\alpha+1}}{q}{n}} \left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;\alpha+n,\beta+n|q)\right] }

Substitution(s): w ⁢ ( x ; α , β | q ) = ( q ⁢ x ; q ) ∞ ( q β + 1 ⁢ x ; q ) ∞ ⁢ x α 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 q-Pochhammer-symbol 𝑞 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛽 1 𝑥 𝑞 superscript 𝑥 𝛼 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)=\frac{\left(qx;% q\right)_{\infty}}{\left(q^{\beta+1}x;q\right)_{\infty}}x^{\alpha}}}}


Generating function

\qHyperrphis ⁢ 01 ⁢ @ ⁢ @ - a ⁢ q ⁢ q ⁢ a ⁢ q ⁢ x ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ x - 1 , 0 ⁢ b ⁢ q ⁢ q ⁢ x ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( b ⁢ q , q ; q ) n ⁢ p n ⁡ ( x ; a , b ; q ) ⁢ t n \qHyperrphis 01 @ @ 𝑎 𝑞 𝑞 𝑎 𝑞 𝑥 𝑡 \qHyperrphis 21 @ @ superscript 𝑥 1 0 𝑏 𝑞 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑏 𝑞 𝑞 𝑞 𝑛 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{0}{1}@@{-}{aq}{q}{aqxt}% \,\qHyperrphis{2}{1}@@{x^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q% ^{\binomial{n}{2}}}{\left(bq,q;q\right)_{n}}p_{n}\!\left(x;a,b;q\right)t^{n}}}} {\displaystyle \qHyperrphis{0}{1}@@{-}{aq}{q}{aqxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{bq,q}{q}{n}}\littleqJacobi{n}@{x}{a}{b}{q}t^n }

Limit relations

Big q-Jacobi polynomial to Little q-Jacobi polynomial

lim c → - ∞ ⁡ P n ⁡ ( c ⁢ q ⁢ x ; a , b , c ; q ) = p n ⁡ ( x ; a , b ; q ) subscript → 𝑐 big-q-Jacobi-polynomial-P 𝑛 𝑐 𝑞 𝑥 𝑎 𝑏 𝑐 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow-\infty}P_{n}\!% \left(cqx;a,b,c;q\right)=p_{n}\!\left(x;a,b;q\right)}}} {\displaystyle \lim_{c\rightarrow -\infty}\bigqJacobi{n}@{cqx}{a}{b}{c}{q}=\littleqJacobi{n}@{x}{a}{b}{q} }

q-Hahn polynomial to Little q-Jacobi polynomial

lim N → ∞ ⁡ Q n ⁡ ( q x - N ; α , β , N ; q ) = p n ⁡ ( q x ; α , β ; q ) subscript → 𝑁 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝑁 𝛼 𝛽 𝑁 𝑞 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(q^{x-N};\alpha,\beta,N;q\right)=p_{n}\!\left(q^{x};\alpha,\beta;q\right)% }}} {\displaystyle \lim _{N\rightarrow\infty} \qHahn{n}@{q^{x-N}}{\alpha}{\beta}{N}{q}=\littleqJacobi{n}@{q^x}{\alpha}{\beta}{q} }

Little q-Jacobi polynomial to Little q-Laguerre / Wall polynomial

p n ⁡ ( x ; a , 0 ; q ) = p n ⁡ ( x ; a ; q ) little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 0 𝑞 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,0;q\right)=p_{n}\!% \left(x;a;q\right)}}} {\displaystyle \littleqJacobi{n}@{x}{a}{0}{q}=\littleqLaguerre{n}@{x}{a}{q} }

Little q-Jacobi polynomial to q-Laguerre polynomial

lim b → - ∞ ⁡ p n ⁡ ( - b - 1 ⁢ q - 1 ⁢ x ; q α , b ; q ) = ( q ; q ) n ( q α + 1 ; q ) n ⁢ L n ( α ) ⁡ ( x ; q ) subscript → 𝑏 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑏 1 superscript 𝑞 1 𝑥 superscript 𝑞 𝛼 𝑏 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{b\rightarrow-\infty}p_{n}\!% \left(-b^{-1}q^{-1}x;q^{\alpha},b;q\right)=\frac{\left(q;q\right)_{n}}{\left(q% ^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)}}} {\displaystyle \lim_{b\rightarrow -\infty}\littleqJacobi{n}@{-b^{-1}q^{-1}x}{q^{\alpha}}{b}{q}= \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q} }

Little q-Jacobi polynomial to q-Bessel polynomial

lim a → 0 ⁡ p n ⁡ ( x ; a , - a - 1 ⁢ q - 1 ⁢ b ; q ) = y n ⁡ ( x ; b ; q ) subscript → 𝑎 0 little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 superscript 𝑎 1 superscript 𝑞 1 𝑏 𝑞 q-Bessel-polynomial-y 𝑛 𝑥 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}p_{n}\!\left(x;% a,-a^{-1}q^{-1}b;q\right)=y_{n}\!\left(x;b;q\right)}}} {\displaystyle \lim_{a\rightarrow 0}\littleqJacobi{n}@{x}{a}{-a^{-1}q^{-1}b}{q}=\qBesselPoly{n}@{x}{b}{q} }

Little q-Jacobi polynomial to Jacobi / Laguerre polynomial

lim q → 1 ⁡ p n ⁡ ( x ; q α , q β ; q ) = P n ( α , β ) ⁡ ( 1 - 2 ⁢ x ) P n ( α , β ) ⁡ ( 1 ) subscript → 𝑞 1 little-q-Jacobi-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}p_{n}\!\left(x;% q^{\alpha},q^{\beta};q\right)=\frac{P^{(\alpha,\beta)}_{n}\left(1-2x\right)}{P% ^{(\alpha,\beta)}_{n}\left(1\right)}}}} {\displaystyle \lim_{q\rightarrow 1}\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}} {\Jacobi{\alpha}{\beta}{n}@{1}} }
lim q → 1 ⁡ p n ⁡ ( 1 2 ⁢ ( 1 - q ) ⁢ x ; q α , - q β ; q ) = L n α ⁡ ( x ) L n α ⁡ ( 0 ) subscript → 𝑞 1 little-q-Jacobi-polynomial-p 𝑛 1 2 1 𝑞 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}p_{n}\!\left(% \textstyle\frac{1}{2}(1-q)x;q^{\alpha},-q^{\beta};q\right)=\frac{L^{\alpha}_{n% }\left(x\right)}{L^{\alpha}_{n}\left(0\right)}}}} {\displaystyle \lim_{q\rightarrow 1}\littleqJacobi{n}@{\textstyle\frac{1}{2}(1-q)x}{q^{\alpha}}{-q^{\beta}}{q} =\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }

Remarks

p n ⁡ ( x ; a , b ; q ) = ( b ⁢ q ; q ) n ( a ⁢ q ; q ) n ⁢ ( - 1 ) n ⁢ b - n ⁢ q - n - \binomial ⁢ n ⁢ 2 ⁢ P n ⁡ ( b ⁢ q ⁢ x ; b , a , 0 ; q ) little-q-Jacobi-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑞 q-Pochhammer-symbol 𝑏 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 superscript 1 𝑛 superscript 𝑏 𝑛 superscript 𝑞 𝑛 \binomial 𝑛 2 big-q-Jacobi-polynomial-P 𝑛 𝑏 𝑞 𝑥 𝑏 𝑎 0 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right)=\frac{% \left(bq;q\right)_{n}}{\left(aq;q\right)_{n}}(-1)^{n}b^{-n}q^{-n-\binomial{n}{% 2}}P_{n}\!\left(bqx;b,a,0;q\right)}}} {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{aq}{q}{n}}(-1)^nb^{-n}q^{-n-\binomial{n}{2}} \bigqJacobi{n}@{bqx}{b}{a}{0}{q} }
M n ⁡ ( q - x ; b , c ; q ) = p n ⁡ ( - c - 1 ⁢ q n ; b , b - 1 ⁢ q - n - x - 1 ; q ) q-Meixner-polynomial-M 𝑛 superscript 𝑞 𝑥 𝑏 𝑐 𝑞 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑐 1 superscript 𝑞 𝑛 𝑏 superscript 𝑏 1 superscript 𝑞 𝑛 𝑥 1 𝑞 {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x};b,c;q\right)=p_% {n}\!\left(-c^{-1}q^{n};b,b^{-1}q^{-n-x-1};q\right)}}} {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\littleqJacobi{n}@{-c^{-1}q^n}{b}{b^{-1}q^{-n-x-1}}{q} }

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