Q-Bessel

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q-Bessel

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qBesselPoly{n}@{x}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},-aq^n}{0}{q}{qx} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qBesselPoly{n}@{x}{a}{q}=\qPochhammer{q^{-n+1}x}{q}{n}\cdot\qHyperrphis{1}{1}@@{q^{-n}}{q^{-n+1}x}{q}{-aq^{n+1}x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qBesselPoly{n}@{x}{a}{q}=\left(-aq^nx\right)^n\cdot\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{-\frac{q^{-n+1}}{a}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{k=0}^{\infty}\frac{a^k}{\qPochhammer{q}{q}{k}}q^{\binomial{k+1}{2}}\qBesselPoly{m}@{q^k}{a}{q}\qBesselPoly{n}@{q^k}{a}{q} {}=\qPochhammer{q}{q}{n}\qPochhammer{-aq^n}{q}{\infty}\frac{a^nq^{\binomial{n+1}{2}}}{(1+aq^{2n})}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a>0}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}}}


Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^{2n-1}\frac{(1-q^n)}{(1+aq^{2n-1})(1+aq^{2n})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1+aq^n)}{(1+aq^{2n})(1+aq^{2n+1})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qBesselPoly{n}@{x}{a}{q}=(-1)^nq^{-\binomial{n}{2}}\qPochhammer{-aq^n}{q}{n}\monicqBesselPoly{n}@@{x}{a}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qBesselPoly{n}@{x}{a}{q}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)=\frac{a^xq^{\binomial{x}{2}}}{\qPochhammer{q}{q}{x}}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)\qBesselPoly{n}@{q^x}{a}{q}=a^n(1-q)^nq^{n(n-1)}\left(\nabla_q\right)^n\left[w(x;aq^{2n};q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)=\frac{a^xq^{\binomial{x}{2}}}{\qPochhammer{q}{q}{x}}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^x} }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{0}{1}@@{-}{0}{q}{-aq^{x+1}t}\,\qHyperrphis{2}{0}@@{q^{-x},0}{-}{q}{q^xt} {}=\sum_{n=0}^{\infty}\frac{\qBesselPoly{n}@{q^x}{a}{q}}{\qPochhammer{q}{q}{n}}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=0,1,2,\ldots}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{t}{q}{\infty}}{\qPochhammer{xt}{q}{\infty}}\,\qHyperrphis{1}{3}@@{xt}{0,0,t}{q}{-aqxt}= \sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\qBesselPoly{n}@{x}{a}{q}t^n }}

Limit relations

Little q-Jacobi polynomial to q-Bessel polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qBesselPoly{n}@{q^x}{a}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[x-n]{n}@{aq^n}{q} }}

Reference

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