q-Bessel

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q-Bessel

Basic hypergeometric representation

y n ⁡ ( x ; a ; q ) = \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , - a ⁢ q n ⁢ 0 ⁢ q ⁢ q ⁢ x q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑎 superscript 𝑞 𝑛 0 𝑞 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=% \qHyperrphis{2}{1}@@{q^{-n},-aq^{n}}{0}{q}{qx}}}} {\displaystyle \qBesselPoly{n}@{x}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},-aq^n}{0}{q}{qx} }
y n ⁡ ( x ; a ; q ) = ( q - n + 1 ⁢ x ; q ) n ⋅ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ q - n ⁢ q - n + 1 ⁢ x ⁢ q - a ⁢ q n + 1 ⁢ x q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 ⋅ q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑥 𝑞 𝑛 \qHyperrphis 11 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 𝑥 𝑞 𝑎 superscript 𝑞 𝑛 1 𝑥 {\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 \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} }
y n ⁡ ( x ; a ; q ) = ( - a ⁢ q n ⁢ x ) n ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - n , x - 1 ⁢ 0 ⁢ q - q - n + 1 a q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 ⋅ superscript 𝑎 superscript 𝑞 𝑛 𝑥 𝑛 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 superscript 𝑥 1 0 𝑞 superscript 𝑞 𝑛 1 𝑎 {\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}}}}} {\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)

∑ k = 0 ∞ a k ( q ; q ) k ⁢ q \binomial ⁢ k + 12 ⁢ y m ⁡ ( q k ; a ; q ) ⁢ y n ⁡ ( q k ; a ; q ) = ( q ; q ) n ⁢ ( - a ⁢ q n ; q ) ∞ ⁢ a n ⁢ q \binomial ⁢ n + 12 ( 1 + a ⁢ q 2 ⁢ n ) ⁢ δ m , n superscript subscript 𝑘 0 superscript 𝑎 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 𝑞 \binomial 𝑘 12 q-Bessel-polynomial-y 𝑚 superscript 𝑞 𝑘 𝑎 𝑞 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑘 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 12 1 𝑎 superscript 𝑞 2 𝑛 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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): a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}}


Recurrence relation

- x ⁢ y n ⁡ ( x ; a ; q ) = A n ⁢ y n + 1 ⁡ ( x ; a ; q ) - ( A n + C n ) ⁢ y n ⁡ ( x ; a ; q ) + C n ⁢ y n - 1 ⁡ ( x ; a ; q ) 𝑥 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 subscript 𝐶 𝑛 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 𝑞 {\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)}}} {\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): C n = a ⁢ q 2 ⁢ n - 1 ⁢ ( 1 - q n ) ( 1 + a ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + a ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 1 𝑎 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{2n-1}\frac{(1-q^{n})}{(1+% aq^{2n-1})(1+aq^{2n})}}}} &
A n = q n ⁢ ( 1 + a ⁢ q n ) ( 1 + a ⁢ q 2 ⁢ n ) ⁢ ( 1 + a ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1+aq^{n})}{(1+aq^% {2n})(1+aq^{2n+1})}}}}


Monic recurrence relation

x ⁢ y ^ n ⁡ ( x ) = y ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ y ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ y ^ n - 1 ⁡ ( x ) 𝑥 q-Bessel-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 q-Bessel-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Bessel-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Bessel-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\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)}}} {\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): C n = a ⁢ q 2 ⁢ n - 1 ⁢ ( 1 - q n ) ( 1 + a ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + a ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 1 𝑎 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{2n-1}\frac{(1-q^{n})}{(1+% aq^{2n-1})(1+aq^{2n})}}}} &
A n = q n ⁢ ( 1 + a ⁢ q n ) ( 1 + a ⁢ q 2 ⁢ n ) ⁢ ( 1 + a ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 𝑎 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}\frac{(1+aq^{n})}{(1+aq^% {2n})(1+aq^{2n+1})}}}}


y n ⁡ ( x ; a ; q ) = ( - 1 ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ ( - a ⁢ q n ; q ) n ⁢ y ^ n ⁡ ( x ) q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 q-Bessel-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 {\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)% }}} {\displaystyle \qBesselPoly{n}@{x}{a}{q}=(-1)^nq^{-\binomial{n}{2}}\qPochhammer{-aq^n}{q}{n}\monicqBesselPoly{n}@@{x}{a}{q} }

q-Difference equation

- q - n ⁢ ( 1 - q n ) ⁢ ( 1 + a ⁢ q n ) ⁢ x ⁢ y ⁢ ( x ) = a ⁢ x ⁢ y ⁢ ( q ⁢ x ) - ( a ⁢ x + 1 - x ) ⁢ y ⁢ ( x ) + ( 1 - x ) ⁢ y ⁢ ( q - 1 ⁢ x ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 𝑥 𝑦 𝑥 𝑎 𝑥 𝑦 𝑞 𝑥 𝑎 𝑥 1 𝑥 𝑦 𝑥 1 𝑥 𝑦 superscript 𝑞 1 𝑥 {\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)}}} {\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): y ⁢ ( x ) = y n ⁡ ( x ; a ; q ) 𝑦 𝑥 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=y_{n}\!\left(x;a;q\right)}}}


Forward shift operator

y n ⁡ ( x ; a ; q ) - y n ⁡ ( q ⁢ x ; a ; q ) = - q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 + a ⁢ q n ) ⁢ x ⁢ y n - 1 ⁡ ( x ; a ⁢ q 2 ; q ) q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 q-Bessel-polynomial-y 𝑛 𝑞 𝑥 𝑎 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 𝑥 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 superscript 𝑞 2 𝑞 {\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 \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} }
𝒟 q ⁢ y n ⁡ ( x ; a ; q ) = - q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 + a ⁢ q n ) 1 - q ⁢ y n - 1 ⁡ ( x ; a ⁢ q 2 ; q ) q-derivative 𝑞 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 𝑞 q-Bessel-polynomial-y 𝑛 1 𝑥 𝑎 superscript 𝑞 2 𝑞 {\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)}}} {\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

a ⁢ q x - 1 ⁢ y n ⁡ ( q x ; a ; q ) - ( 1 - q x ) ⁢ y n ⁡ ( q x - 1 ⁢ x ; a ; q ) = - y n + 1 ⁡ ( q x ; a ⁢ q - 2 ; q ) 𝑎 superscript 𝑞 𝑥 1 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 1 superscript 𝑞 𝑥 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 1 𝑥 𝑎 𝑞 q-Bessel-polynomial-y 𝑛 1 superscript 𝑞 𝑥 𝑎 superscript 𝑞 2 𝑞 {\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 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} }
∇ ⁡ [ w ⁢ ( x ; a ; q ) ⁢ y n ⁡ ( q x ; a ; q ) ] ∇ ⁡ q x = q 2 a ⁢ ( 1 - q ) ⁢ w ⁢ ( x ; a ⁢ q - 2 ; q ) ⁢ y n + 1 ⁡ ( q x ; a ⁢ q - 2 ; q ) ∇ 𝑤 𝑥 𝑎 𝑞 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 ∇ superscript 𝑞 𝑥 superscript 𝑞 2 𝑎 1 𝑞 𝑤 𝑥 𝑎 superscript 𝑞 2 𝑞 q-Bessel-polynomial-y 𝑛 1 superscript 𝑞 𝑥 𝑎 superscript 𝑞 2 𝑞 {\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)}}} {\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): w ⁢ ( x ; a ; q ) = a x ⁢ q \binomial ⁢ x ⁢ 2 ( q ; q ) x 𝑤 𝑥 𝑎 𝑞 superscript 𝑎 𝑥 superscript 𝑞 \binomial 𝑥 2 q-Pochhammer-symbol 𝑞 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x}% {2}}}{\left(q;q\right)_{x}}}}}


Rodrigues-type formula

w ⁢ ( x ; a ; q ) ⁢ y n ⁡ ( q x ; a ; q ) = a n ⁢ ( 1 - q ) n ⁢ q n ⁢ ( n - 1 ) ⁢ ( ∇ q ) n ⁢ [ w ⁢ ( x ; a ⁢ q 2 ⁢ n ; q ) ] 𝑤 𝑥 𝑎 𝑞 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 superscript 𝑎 𝑛 superscript 1 𝑞 𝑛 superscript 𝑞 𝑛 𝑛 1 superscript subscript ∇ 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝑎 superscript 𝑞 2 𝑛 𝑞 {\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]}}} {\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): w ⁢ ( x ; a ; q ) = a x ⁢ q \binomial ⁢ x ⁢ 2 ( q ; q ) x 𝑤 𝑥 𝑎 𝑞 superscript 𝑎 𝑥 superscript 𝑞 \binomial 𝑥 2 q-Pochhammer-symbol 𝑞 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x}% {2}}}{\left(q;q\right)_{x}}}}}


∇ q := ∇ ∇ ⁡ q x assign subscript ∇ 𝑞 ∇ ∇ superscript 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{% x}}}}} {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^x} }

Generating functions

\qHyperrphis ⁢ 01 ⁢ @ ⁢ @ - 0 ⁢ q - a ⁢ q x + 1 ⁢ t ⁢ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q - x , 0 - q ⁢ q x ⁢ t = ∑ n = 0 ∞ y n ⁡ ( q x ; a ; q ) ( q ; q ) n ⁢ t n \qHyperrphis 01 @ @ 0 𝑞 𝑎 superscript 𝑞 𝑥 1 𝑡 \qHyperrphis 20 @ @ superscript 𝑞 𝑥 0 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\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}}}} {\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): x = 0 , 1 , 2 , … 𝑥 0 1 2 … {\displaystyle{\displaystyle{\displaystyle x=0,1,2,\ldots}}}


( t ; q ) ∞ ( x ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 13 ⁢ @ ⁢ @ ⁢ x ⁢ t ⁢ 0 , 0 , t ⁢ q - a ⁢ q ⁢ x ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q ; q ) n ⁢ y n ⁡ ( x ; a ; q ) ⁢ t n q-Pochhammer-symbol 𝑡 𝑞 q-Pochhammer-symbol 𝑥 𝑡 𝑞 \qHyperrphis 13 @ @ 𝑥 𝑡 0 0 𝑡 𝑞 𝑎 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Bessel-polynomial-y 𝑛 𝑥 𝑎 𝑞 superscript 𝑡 𝑛 {\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}}}} {\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

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} }

q-Krawtchouk polynomial to q-Bessel polynomial

lim N → ∞ ⁡ K n ⁡ ( q x - N ; p , N ; q ) = y n ⁡ ( q x ; p ; q ) subscript → 𝑁 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑁 𝑝 𝑁 𝑞 q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑝 𝑞 {\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)}}} {\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

lim a → ∞ ⁡ y n ⁡ ( a - 1 ⁢ x ; a ; q ) = ( q ; q ) n ⁢ S n ⁡ ( x ; q ) subscript → 𝑎 q-Bessel-polynomial-y 𝑛 superscript 𝑎 1 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 {\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)}}} {\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

lim q → 1 ⁡ y n ⁡ ( - 1 2 ⁢ ( 1 - q ) - 1 ⁢ x ; - q a + 1 ; q ) = y n ⁡ ( x ; a ) subscript → 𝑞 1 q-Bessel-polynomial-y 𝑛 1 2 superscript 1 𝑞 1 𝑥 superscript 𝑞 𝑎 1 𝑞 Bessel-polynomial-y 𝑛 𝑥 𝑎 {\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)}}} {\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

lim q → 1 ⁡ y n ⁡ ( q x ; a ( 1 - q ) ; q ) ( q - 1 ) n = a n ⁢ C n ⁡ ( x ; a ) subscript → 𝑞 1 fragments Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 fragments a fragments ( 1 q ; q ) superscript 𝑞 1 𝑛 superscript 𝑎 𝑛 Charlier-polynomial-C 𝑛 𝑥 𝑎 {\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)}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\BesselPoly{n}@{q^x}{a(1-q};q)}{(q-1)^n}=a^n\Charlier{n}@{x}{a} }

Remark

y n ⁡ ( q x ; a ; q ) ( q ; q ) n = L n ( x - n ) ⁡ ( a ⁢ q n ; q ) q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Laguerre-polynomial-L 𝑥 𝑛 𝑛 𝑎 superscript 𝑞 𝑛 𝑞 {\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)}}} {\displaystyle \frac{\qBesselPoly{n}@{q^x}{a}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[x-n]{n}@{aq^n}{q} }

Reference

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