Little q-Jacobi: Special case

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Little q-Jacobi: Special case

Little q-Legendre

Basic hypergeometric representation

p n ⁡ ( x | q ) = \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , q n + 1 ⁢ q ⁢ q ⁢ q ⁢ x little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 𝑞 𝑞 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=% \qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx}}}} {\displaystyle \littleqLegendre{n}@{x}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx} }

Orthogonality relation(s)

∫ 0 1 p m ⁡ ( x | q ) ⁢ p n ⁡ ( x | q ) ⁢ d q ⁢ x = ( 1 - q ) ⁢ ∑ k = 0 ∞ q k ⁢ p m ⁡ ( q k | q ) ⁢ p n ⁡ ( q k | q ) superscript subscript 0 1 little-q-Legendre-polynomial-p 𝑚 𝑥 𝑞 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝑑 𝑞 𝑥 1 𝑞 superscript subscript 𝑘 0 superscript 𝑞 𝑘 little-q-Legendre-polynomial-p 𝑚 superscript 𝑞 𝑘 𝑞 little-q-Legendre-polynomial-p 𝑛 superscript 𝑞 𝑘 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{1}p_{m}\!\left(x|q\right)p% _{n}\!\left(x|q\right)\,d_{q}x=(1-q)\sum_{k=0}^{\infty}q^{k}p_{m}\!\left(q^{k}% |q\right)p_{n}\!\left(q^{k}|q\right)}}} {\displaystyle \int_0^1\littleqLegendre{m}@{x}{q}\littleqLegendre{n}@{x}{q}\,d_qx=(1-q)\sum_{k=0}^{\infty}q^k\littleqLegendre{m}@{q^k}{q}\littleqLegendre{n}@{q^k}{q} }
∫ 0 1 p m ⁡ ( x | q ) ⁢ p n ⁡ ( x | q ) ⁢ d q ⁢ x = ( 1 - q ) ⁢ q n ( 1 - q 2 ⁢ n + 1 ) ⁢ δ m , n superscript subscript 0 1 little-q-Legendre-polynomial-p 𝑚 𝑥 𝑞 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝑑 𝑞 𝑥 1 𝑞 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 1 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{1}p_{m}\!\left(x|q\right)p% _{n}\!\left(x|q\right)\,d_{q}x=\frac{(1-q)q^{n}}{(1-q^{2n+1})}\,\delta_{m,n}}}} {\displaystyle \int_0^1\littleqLegendre{m}@{x}{q}\littleqLegendre{n}@{x}{q}\,d_qx=\frac{(1-q)q^n}{(1-q^{2n+1})}\,\Kronecker{m}{n} }

Recurrence relation

- x ⁢ p n ⁡ ( x | q ) = A n ⁢ p n + 1 ⁡ ( x | q ) - ( A n + C n ) ⁢ p n ⁡ ( x | q ) + C n ⁢ p n - 1 ⁡ ( x | q ) 𝑥 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝐴 𝑛 little-q-Legendre-polynomial-p 𝑛 1 𝑥 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝐶 𝑛 little-q-Legendre-polynomial-p 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x|q\right)=A_{n}p_{n+% 1}\!\left(x|q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x|q\right)+C_{n}p_{n% -1}\!\left(x|q\right)}}} {\displaystyle -x\littleqLegendre{n}@{x}{q}=A_n\littleqLegendre{n+1}@{x}{q}-\left(A_n+C_n\right)\littleqLegendre{n}@{x}{q}+C_n\littleqLegendre{n-1}@{x}{q} }

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


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-Legendre-polynomial-monic-p 𝑛 𝑥 𝑞 little-q-Legendre-polynomial-monic-p 𝑛 1 𝑥 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Legendre-polynomial-monic-p 𝑛 𝑥 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 little-q-Legendre-polynomial-monic-p 𝑛 1 𝑥 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\moniclittleqLegendre{n}@@{x}{q}=\moniclittleqLegendre{n+1}@@{x}{q}+(A_n+C_n)\moniclittleqLegendre{n}@@{x}{q}+A_{n-1}C_n\moniclittleqLegendre{n-1}@@{x}{q} }

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


p n ⁡ ( x | q ) = ( - 1 ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ ( q n + 1 ; q ) n ( q ; q ) n ⁢ p ^ n ⁡ ( x ) little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 little-q-Legendre-polynomial-monic-p 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=\frac{(-1)^% {n}q^{-\binomial{n}{2}}\left(q^{n+1};q\right)_{n}}{\left(q;q\right)_{n}}{% \widehat{p}}_{n}\!\left(x\right)}}} {\displaystyle \littleqLegendre{n}@{x}{q}=\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\moniclittleqLegendre{n}@@{x}{q} }

q-Difference equation

q - n ⁢ ( 1 - q n ) ⁢ ( 1 - 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 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-q^{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-q^{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 ) = q ⁢ x - 1 𝐵 𝑥 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=qx-1}}} &

y ⁢ ( x ) = p n ⁡ ( x | q ) 𝑦 𝑥 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x|q\right)}}}


Rodrigues-type formula

p n ⁡ ( x | q ) = q \binomial ⁢ n ⁢ 2 ⁢ ( 1 - q ) n ( q ; q ) n ⁢ ( 𝒟 q - 1 ) n ⁢ [ ( q ⁢ x ; q ) n ⁢ x n ] little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 superscript 𝑞 \binomial 𝑛 2 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript 𝒟 superscript 𝑞 1 𝑛 delimited-[] q-Pochhammer-symbol 𝑞 𝑥 𝑞 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=\frac{q^{% \binomial{n}{2}}(1-q)^{n}}{\left(q;q\right)_{n}}\left(\mathcal{D}_{q^{-1}}% \right)^{n}\left[\left(qx;q\right)_{n}x^{n}\right]}}} {\displaystyle \littleqLegendre{n}@{x}{q}=\frac{q^{\binomial{n}{2}}(1-q)^n}{\qPochhammer{q}{q}{n}} \left(\mathcal{D}_{q^{-1}}\right)^n\left[\qPochhammer{qx}{q}{n}x^n\right] }

Generating function

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

Limit relation

Little q-Legendre polynomial to Legendre / Spherical polynomial

lim q → 1 ⁡ p n ⁡ ( x | q ) = \LegendrePoly ⁢ n ⁢ @ ⁢ 1 - 2 ⁢ x subscript → 𝑞 1 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 \LegendrePoly 𝑛 @ 1 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}p_{n}\!\left(x|% q\right)=\LegendrePoly{n}@{1-2x}}}} {\displaystyle \lim_{q\rightarrow 1}\littleqLegendre{n}@{x}{q}=\LegendrePoly{n}@{1-2x} }

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