Q-Bessel

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Basic hypergeometric representation

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

Orthogonality relation(s)

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

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-x\qBesselPoly{n}@{x}{a}{q}=A_{n}\qBesselPoly{n+1}@{x}{a}{q}-(A_{n}+C_{n})\qBesselPoly{n}@{x}{a}{q}+C_{n}\qBesselPoly{n-1}@{x}{a}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle x\monicqBesselPoly{n}@@{x}{a}{q}=\monicqBesselPoly{n+1}@@{x}{a}{q}+(A_{n}+C_{n})\monicqBesselPoly{n}@@{x}{a}{q}+A_{n-1}C_{n}\monicqBesselPoly{n-1}@@{x}{a}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qBesselPoly{n}@{x}{a}{q}=(-1)^{n}q^{-\binomial{n}{2}}\qPochhammer{-aq^{n}}{q}{n}\monicqBesselPoly{n}@@{x}{a}{q}}}}$

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): ${\displaystyle{\displaystyle{\displaystyle y(x)=\qBesselPoly{n}@{x}{a}{q}}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\qBesselPoly{n}@{x}{a}{q}-\qBesselPoly{n}@{qx}{a}{q}=-q^{-n+1}(1-q^{n})(1+aq^{n})x\qBesselPoly{n-1}@{x}{aq^{2}}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\qderiv{q}\qBesselPoly{n}@{x}{a}{q}=-\frac{q^{-n+1}(1-q^{n})(1+aq^{n})}{1-q}\qBesselPoly{n-1}@{x}{aq^{2}}{q}}}}$

Backward shift operator

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

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

Rodrigues-type formula

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

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

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

Generating functions

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

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

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

Limit relations

Little q-Jacobi polynomial to q-Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}\littleqJacobi{n}@{x}{a}{-a^{-1}q^{-1}b}{q}=\qBesselPoly{n}@{x}{b}{q}}}}$

q-Krawtchouk polynomial to q-Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\qKrawtchouk{n}@{q^{x-N}}{p}{N}{q}=\qBesselPoly{n}@{q^{x}}{p}{q}}}}$

q-Bessel polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}\qBesselPoly{n}@{a^{-1}x}{a}{q}=\qPochhammer{q}{q}{n}\StieltjesWigert{n}@{x}{q}}}}$

q-Bessel polynomial to Bessel polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\qBesselPoly{n}@{-\textstyle\frac{1}{2}(1-q)^{-1}x}{-q^{a+1}}{q}=\BesselPoly{n}@{x}{a}}}}$

q-Bessel polynomial to Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{\BesselPoly{n}@{q^{x}}{a(1-q};q)}{(q-1)^{n}}=a^{n}\Charlier{n}@{x}{a}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\frac{\qBesselPoly{n}@{q^{x}}{a}{q}}{\left(q;q\right)_{n}}=\qLaguerre[x-n]{n}@{aq^{n}}{q}}}}$

Reference

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