q-Bessel

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=\qHyperrphis{2}{1}@@{q^{-n},-aq^{n}}{0}{q}{qx}}}}$
${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=\left(q^{-n+1}x;q\right)_{n}\cdot\qHyperrphis{1}{1}@@{q^{-n}}{q^{-n+1}x}{q}{-aq^{n+1}x}}}}$
${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=\left(-aq^{n}x\right)^{n}\cdot\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{-\frac{q^{-n+1}}{a}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{a^{k}}{\left(q;q\right)_{k}}q^{\binomial{k+1}{2}}y_{m}\!\left(q^{k};a;q\right)y_{n}\!\left(q^{k};a;q\right){}=\left(q;q\right)_{n}\left(-aq^{n};q\right)_{\infty}\frac{a^{n}q^{\binomial{n+1}{2}}}{(1+aq^{2n})}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-xy_{n}\!\left(x;a;q\right)=A_{n}y_{n+1}\!\left(x;a;q\right)-(A_{n}+C_{n})y_{n}\!\left(x;a;q\right)+C_{n}y_{n-1}\!\left(x;a;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{2n-1}\frac{(1-q^{n})}{(1+aq^{2n-1})(1+aq^{2n})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1+aq^{n})}{(1+aq^{2n})(1+aq^{2n+1})}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{y}}_{n}\!\left(x\right)={\widehat{y}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}){\widehat{y}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{y}}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{2n-1}\frac{(1-q^{n})}{(1+aq^{2n-1})(1+aq^{2n})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1+aq^{n})}{(1+aq^{2n})(1+aq^{2n+1})}}}}$

${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=(-1)^{n}q^{-\binomial{n}{2}}\left(-aq^{n};q\right)_{n}{\widehat{y}}_{n}\!\left(x\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle-q^{-n}(1-q^{n})(1+aq^{n})xy(x){}=axy(qx)-(ax+1-x)y(x)+(1-x)y(q^{-1}x)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=y_{n}\!\left(x;a;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)-y_{n}\!\left(qx;a;q\right)=-q^{-n+1}(1-q^{n})(1+aq^{n})xy_{n-1}\!\left(x;aq^{2};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}y_{n}\!\left(x;a;q\right)=-\frac{q^{-n+1}(1-q^{n})(1+aq^{n})}{1-q}y_{n-1}\!\left(x;aq^{2};q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle aq^{x-1}y_{n}\!\left(q^{x};a;q\right)-(1-q^{x})y_{n}\!\left(q^{x-1}x;a;q\right)=-y_{n+1}\!\left(q^{x};aq^{-2};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;a;q)y_{n}\!\left(q^{x};a;q\right)\right]}{\nabla q^{x}}=\frac{q^{2}}{a(1-q)}w(x;aq^{-2};q)y_{n+1}\!\left(q^{x};aq^{-2};q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x}{2}}}{\left(q;q\right)_{x}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)y_{n}\!\left(q^{x};a;q\right)=a^{n}(1-q)^{n}q^{n(n-1)}\left(\nabla_{q}\right)^{n}\left[w(x;aq^{2n};q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x}{2}}}{\left(q;q\right)_{x}}}}}$

${\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{x}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{0}{1}@@{-}{0}{q}{-aq^{x+1}t}\,\qHyperrphis{2}{0}@@{q^{-x},0}{-}{q}{q^{x}t}{}=\sum_{n=0}^{\infty}\frac{y_{n}\!\left(q^{x};a;q\right)}{\left(q;q\right)_{n}}t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{\left(t;q\right)_{\infty}}{\left(xt;q\right)_{\infty}}\,\qHyperrphis{1}{3}@@{xt}{0,0,t}{q}{-aqxt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}y_{n}\!\left(x;a;q\right)t^{n}}}}$

Limit relations

Little q-Jacobi polynomial to q-Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}p_{n}\!\left(x;a,-a^{-1}q^{-1}b;q\right)=y_{n}\!\left(x;b;q\right)}}}$

q-Krawtchouk polynomial to q-Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!\left(q^{x-N};p,N;q\right)=y_{n}\!\left(q^{x};p;q\right)}}}$

q-Bessel polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}y_{n}\!\left(a^{-1}x;a;q\right)=\left(q;q\right)_{n}S_{n}\!\left(x;q\right)}}}$

q-Bessel polynomial to Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}y_{n}\!\left(-\textstyle\frac{1}{2}(1-q)^{-1}x;-q^{a+1};q\right)=y_{n}\!\left(x;a\right)}}}$

q-Bessel polynomial to Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{y_{n}\!\left(q^{x};a(1-q\right);q)}{(q-1)^{n}}=a^{n}C_{n}\!\left(x;a\right)}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\frac{y_{n}\!\left(q^{x};a;q\right)}{\left(q;q\right)_{n}}=L^{(x-n)}_{n}\!\left(aq^{n};q\right)}}}$

Reference

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