M n â¡ ( q - x ; b , c ; q ) = \qHyperrphis ⢠21 ⢠@ ⢠@ ⢠q - n , q - x ⢠b ⢠q ⢠q - q n + 1 c q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð \qHyperrphis 21 @ @ superscript ð ð superscript ð ð¥ ð ð ð superscript ð ð 1 ð {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x};b,c;q\right)=% \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{bq}{q}{-\frac{q^{n+1}}{c}}}}} {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{bq}{q}{-\frac{q^{n+1}}{c}} }
â x = 0 â ( b ⢠q ; q ) x ( q , - b ⢠c ⢠q ; q ) x ⢠c x ⢠q \binomial ⢠x ⢠2 ⢠M m â¡ ( q - x ; b , c ; q ) ⢠M n â¡ ( q - x ; b , c ; q ) = ( - c ; q ) â ( - b ⢠c ⢠q ; q ) â ⢠( q , - c - 1 ⢠q ; q ) n ( b ⢠q ; q ) n ⢠q - n ⢠δ m , n superscript subscript ð¥ 0 q-Pochhammer-symbol ð ð ð ð¥ q-Pochhammer-symbol ð ð ð ð ð ð¥ superscript ð ð¥ superscript ð \binomial ð¥ 2 q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð q-Pochhammer-symbol ð ð q-Pochhammer-symbol ð ð ð ð q-Pochhammer-symbol ð superscript ð 1 ð ð ð q-Pochhammer-symbol ð ð ð ð superscript ð ð Kronecker-delta ð ð {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{\left(bq;q% \right)_{x}}{\left(q,-bcq;q\right)_{x}}c^{x}q^{\binomial{x}{2}}M_{m}\!\left(q^% {-x};b,c;q\right)M_{n}\!\left(q^{-x};b,c;q\right){}=\frac{\left(-c;q\right)_{% \infty}}{\left(-bcq;q\right)_{\infty}}\frac{\left(q,-c^{-1}q;q\right)_{n}}{% \left(bq;q\right)_{n}}q^{-n}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^{\infty}\frac{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\binomial{x}{2}}\qMeixner{m}@{q^{-x}}{b}{c}{q}\qMeixner{n}@{q^{-x}}{b}{c}{q} {}=\frac{\qPochhammer{-c}{q}{\infty}}{\qPochhammer{-bcq}{q}{\infty}}\frac{\qPochhammer{q,-c^{-1}q}{q}{n}}{\qPochhammer{bq}{q}{n}}q^{-n}\,\Kronecker{m}{n} }
q 2 ⢠n + 1 ⢠( 1 - q - x ) ⢠M n â¡ ( q - x ) = c ⢠( 1 - b ⢠q n + 1 ) ⢠M n + 1 â¡ ( q - x ) - [ c ⢠( 1 - b ⢠q n + 1 ) + q ⢠( 1 - q n ) ⢠( c + q n ) ] ⢠M n â¡ ( q - x ) + q ⢠( 1 - q n ) ⢠( c + q n ) ⢠M n - 1 â¡ ( q - x ) superscript ð 2 ð 1 1 superscript ð ð¥ q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð ð 1 ð superscript ð ð 1 q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð ð ð delimited-[] ð 1 ð superscript ð ð 1 ð 1 superscript ð ð ð superscript ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð ð 1 superscript ð ð ð superscript ð ð q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle q^{2n+1}(1-q^{-x})M_{n}\!\left(q^{-% x}\right){}=c(1-bq^{n+1})M_{n+1}\!\left(q^{-x}\right){}-\left[c(1-bq^{n+1})+q(% 1-q^{n})(c+q^{n})\right]M_{n}\!\left(q^{-x}\right){}+q(1-q^{n})(c+q^{n})M_{n-1% }\!\left(q^{-x}\right)}}} {\displaystyle q^{2n+1}(1-q^{-x})\qMeixner{n}@@{q^{-x}}{b}{c}{q} {}=c(1-bq^{n+1})\qMeixner{n+1}@@{q^{-x}}{b}{c}{q} {}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]\qMeixner{n}@@{q^{-x}}{b}{c}{q} {}+q(1-q^n)(c+q^n)\qMeixner{n-1}@@{q^{-x}}{b}{c}{q} } M n â¡ ( q - x ) := M n â¡ ( q - x ; b , c ; q ) assign q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x}\right):=M_{n}\!% \left(q^{-x};b,c;q\right)}}} {\displaystyle \qMeixner{n}@@{q^{-x}}{b}{c}{q}:=\qMeixner{n}@{q^{-x}}{b}{c}{q} }
x ⢠M ^ n â¡ ( x ) = M ^ n + 1 â¡ ( x ) + [ 1 + q - 2 ⢠n - 1 ⢠{ c ⢠( 1 - b ⢠q n + 1 ) + q ⢠( 1 - q n ) ⢠( c + q n ) } ] ⢠M ^ n â¡ ( x ) + c ⢠q - 4 ⢠n + 1 ⢠( 1 - q n ) ⢠( 1 - b ⢠q n ) ⢠( c + q n ) ⢠M ^ n - 1 â¡ ( x ) ð¥ q-Meixner-polynomial-monic-M ð ð¥ ð ð ð q-Meixner-polynomial-monic-M ð 1 ð¥ ð ð ð delimited-[] 1 superscript ð 2 ð 1 ð 1 ð superscript ð ð 1 ð 1 superscript ð ð ð superscript ð ð q-Meixner-polynomial-monic-M ð ð¥ ð ð ð ð superscript ð 4 ð 1 1 superscript ð ð 1 ð superscript ð ð ð superscript ð ð q-Meixner-polynomial-monic-M ð 1 ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle x{\widehat{M}}_{n}\!\left(x\right)=% {\widehat{M}}_{n+1}\!\left(x\right)+\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-% q^{n})(c+q^{n})\right\}\right]{\widehat{M}}_{n}\!\left(x\right){}+cq^{-4n+1}(1% -q^{n})(1-bq^{n})(c+q^{n}){\widehat{M}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicqMeixner{n}@@{x}{b}{c}{q}=\monicqMeixner{n+1}@@{x}{b}{c}{q}+ \left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]\monicqMeixner{n}@@{x}{b}{c}{q} {}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)\monicqMeixner{n-1}@@{x}{b}{c}{q} } M n â¡ ( q - x ; b , c ; q ) = ( - 1 ) n ⢠q n 2 ( b ⢠q ; q ) n ⢠c n ⢠M ^ n â¡ ( q - x ) q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð superscript 1 ð superscript ð superscript ð 2 q-Pochhammer-symbol ð ð ð ð superscript ð ð q-Meixner-polynomial-monic-M ð superscript ð ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x};b,c;q\right)=% \frac{(-1)^{n}q^{n^{2}}}{\left(bq;q\right)_{n}c^{n}}{\widehat{M}}_{n}\!\left(q% ^{-x}\right)}}} {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\frac{(-1)^nq^{n^2}}{\qPochhammer{bq}{q}{n}c^n}\monicqMeixner{n}@@{q^{-x}}{b}{c}{q} }
- ( 1 - q n ) ⢠y ⢠( x ) = B ⢠( x ) ⢠y ⢠( x + 1 ) - [ B ⢠( x ) + D ⢠( x ) ] ⢠y ⢠( x ) + D ⢠( x ) ⢠y ⢠( x - 1 ) 1 superscript ð ð ð¦ ð¥ ðµ ð¥ ð¦ ð¥ 1 delimited-[] ðµ ð¥ ð· ð¥ ð¦ ð¥ ð· ð¥ ð¦ ð¥ 1 {\displaystyle{\displaystyle{\displaystyle-(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)% +D(x)\right]y(x)+D(x)y(x-1)}}} {\displaystyle -(1-q^n)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1) }
B ⢠( x ) = c ⢠q x ⢠( 1 - b ⢠q x + 1 ) ðµ ð¥ ð superscript ð ð¥ 1 ð superscript ð ð¥ 1 {\displaystyle{\displaystyle{\displaystyle B(x)=cq^{x}(1-bq^{x+1})}}} &
M n â¡ ( q - x - 1 ; b , c ; q ) - M n â¡ ( q - x ; b , c ; q ) = - q - x ⢠( 1 - q n ) c ⢠( 1 - b ⢠q ) ⢠M n - 1 â¡ ( q - x ; b ⢠q , c ⢠q - 1 ; q ) q-Meixner-polynomial-M ð superscript ð ð¥ 1 ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð superscript ð ð¥ 1 superscript ð ð ð 1 ð ð q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð ð ð superscript ð 1 ð {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x-1};b,c;q\right)-% M_{n}\!\left(q^{-x};b,c;q\right){}=-\frac{q^{-x}(1-q^{n})}{c(1-bq)}M_{n-1}\!% \left(q^{-x};bq,cq^{-1};q\right)}}} {\displaystyle \qMeixner{n}@{q^{-x-1}}{b}{c}{q}-\qMeixner{n}@{q^{-x}}{b}{c}{q} {}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}\qMeixner{n-1}@{q^{-x}}{bq}{cq^{-1}}{q} } Π⢠M n â¡ ( q - x ; b , c ; q ) Π⢠q - x = - q ⢠( 1 - q n ) c ⢠( 1 - q ) ⢠( 1 - b ⢠q ) ⢠M n - 1 â¡ ( q - x ; b ⢠q , c ⢠q - 1 ; q ) Î q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð Î superscript ð ð¥ ð 1 superscript ð ð ð 1 ð 1 ð ð q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð ð ð superscript ð 1 ð {\displaystyle{\displaystyle{\displaystyle\frac{\Delta M_{n}\!\left(q^{-x};b,c% ;q\right)}{\Delta q^{-x}}=-\frac{q(1-q^{n})}{c(1-q)(1-bq)}M_{n-1}\!\left(q^{-x% };bq,cq^{-1};q\right)}}} {\displaystyle \frac{\Delta \qMeixner{n}@{q^{-x}}{b}{c}{q}}{\Delta q^{-x}} =-\frac{q(1-q^n)}{c(1-q)(1-bq)}\qMeixner{n-1}@{q^{-x}}{bq}{cq^{-1}}{q} }
c ⢠q x ⢠( 1 - b ⢠q x ) ⢠M n â¡ ( q - x ; b , c ; q ) - ( 1 - q x ) ⢠( 1 + b ⢠c ⢠q x ) ⢠M n â¡ ( q - x + 1 ; b , c ; q ) = c ⢠q x ⢠( 1 - b ) ⢠M n + 1 â¡ ( q - x ; b ⢠q - 1 , c ⢠q ; q ) ð superscript ð ð¥ 1 ð superscript ð ð¥ q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð 1 superscript ð ð¥ 1 ð ð superscript ð ð¥ q-Meixner-polynomial-M ð superscript ð ð¥ 1 ð ð ð ð superscript ð ð¥ 1 ð q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð superscript ð 1 ð ð ð {\displaystyle{\displaystyle{\displaystyle cq^{x}(1-bq^{x})M_{n}\!\left(q^{-x}% ;b,c;q\right)-(1-q^{x})(1+bcq^{x})M_{n}\!\left(q^{-x+1};b,c;q\right){}=cq^{x}(% 1-b)M_{n+1}\!\left(q^{-x};bq^{-1},cq;q\right)}}} {\displaystyle cq^x(1-bq^x)\qMeixner{n}@{q^{-x}}{b}{c}{q}-(1-q^x)(1+bcq^x)\qMeixner{n}@{q^{-x+1}}{b}{c}{q} {}=cq^x(1-b)\qMeixner{n+1}@{q^{-x}}{bq^{-1}}{cq}{q} } â â¡ [ w ⢠( x ; b , c ; q ) ⢠M n â¡ ( q - x ; b , c ; q ) ] â â¡ q - x = 1 1 - q ⢠w ⢠( x ; b ⢠q - 1 , c ⢠q ; q ) ⢠M n + 1 â¡ ( q - x ; b ⢠q - 1 , c ⢠q ; q ) â ð¤ ð¥ ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð â superscript ð ð¥ 1 1 ð ð¤ ð¥ ð superscript ð 1 ð ð ð q-Meixner-polynomial-M ð 1 superscript ð ð¥ ð superscript ð 1 ð ð ð {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;b,c;q)M_{n}\!% \left(q^{-x};b,c;q\right)\right]}{\nabla q^{-x}}{}=\frac{1}{1-q}w(x;bq^{-1},cq% ;q)M_{n+1}\!\left(q^{-x};bq^{-1},cq;q\right)}}} {\displaystyle \frac{\nabla\left[w(x;b,c;q)\qMeixner{n}@{q^{-x}}{b}{c}{q}\right]}{\nabla q^{-x}} {}=\frac{1}{1-q}w(x;bq^{-1},cq;q)\qMeixner{n+1}@{q^{-x}}{bq^{-1}}{cq}{q} }
w ⢠( x ; b , c ; q ) ⢠M n â¡ ( q - x ; b , c ; q ) = ( 1 - q ) n ⢠( â q ) n ⢠[ w ⢠( x ; b ⢠q n , c ⢠q - n ; q ) ] ð¤ ð¥ ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð superscript 1 ð ð superscript subscript â ð ð delimited-[] ð¤ ð¥ ð superscript ð ð ð superscript ð ð ð {\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)M_{n}\!\left(q^{-x};b,c;q% \right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;bq^{n},cq^{-n};q)\right]% }}} {\displaystyle w(x;b,c;q)\qMeixner{n}@{q^{-x}}{b}{c}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right] }
â q := â â â¡ q - x assign subscript â ð â â superscript ð ð¥ {\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{% -x}}}}} {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^{-x}} }
1 ( t ; q ) â ⢠\qHyperrphis ⢠11 ⢠@ ⢠@ ⢠q - x ⢠b ⢠q ⢠q - c - 1 ⢠q ⢠t = â n = 0 â M n â¡ ( q - x ; b , c ; q ) ( q ; q ) n ⢠t n 1 q-Pochhammer-symbol ð¡ ð \qHyperrphis 11 @ @ superscript ð ð¥ ð ð ð superscript ð 1 ð ð¡ superscript subscript ð 0 q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð q-Pochhammer-symbol ð ð ð superscript ð¡ ð {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}% \,\qHyperrphis{1}{1}@@{q^{-x}}{bq}{q}{-c^{-1}qt}=\sum_{n=0}^{\infty}\frac{M_{n% }\!\left(q^{-x};b,c;q\right)}{\left(q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{1}@@{q^{-x}}{bq}{q}{-c^{-1}qt} =\sum_{n=0}^{\infty}\frac{\qMeixner{n}@{q^{-x}}{b}{c}{q}}{\qPochhammer{q}{q}{n}}t^n } 1 ( t ; q ) â ⢠\qHyperrphis ⢠11 ⢠@ ⢠@ - b - 1 ⢠c - 1 ⢠q - x - c - 1 ⢠q ⢠q ⢠b ⢠q ⢠t = â n = 0 â ( b ⢠q ; q ) n ( - c - 1 ⢠q , q ; q ) n ⢠M n â¡ ( q - x ; b , c ; q ) ⢠t n 1 q-Pochhammer-symbol ð¡ ð \qHyperrphis 11 @ @ superscript ð 1 superscript ð 1 superscript ð ð¥ superscript ð 1 ð ð ð ð ð¡ superscript subscript ð 0 q-Pochhammer-symbol ð ð ð ð q-Pochhammer-symbol superscript ð 1 ð ð ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð superscript ð¡ ð {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}% \,\qHyperrphis{1}{1}@@{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{q}{bqt}{}=\sum_{n=0}^{% \infty}\frac{\left(bq;q\right)_{n}}{\left(-c^{-1}q,q;q\right)_{n}}M_{n}\!\left% (q^{-x};b,c;q\right)t^{n}}}} {\displaystyle \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{1}@@{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{q}{bqt} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{-c^{-1}q,q}{q}{n}}\qMeixner{n}@{q^{-x}}{b}{c}{q}t^n }
lim c â - â â¡ \bigqJacobi ⢠n ⢠@ ⢠q - x ⢠a - a - 1 ⢠c ⢠d - 1 ⢠c ⢠q = \qMeixner ⢠n ⢠@ ⢠q - x ⢠a ⢠d ⢠q subscript â ð \bigqJacobi ð @ superscript ð ð¥ ð superscript ð 1 ð superscript ð 1 ð ð \qMeixner ð @ superscript ð ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow-\infty}% \bigqJacobi{n}@{q^{-x}}{a}{-a^{-1}cd^{-1}}{c}{q}=\qMeixner{n}@{q^{-x}}{a}{d}{q% }}}} {\displaystyle \lim_{c\rightarrow -\infty}\bigqJacobi{n}@{q^{-x}}{a}{-a^{-1}cd^{-1}}{c}{q}=\qMeixner{n}@{q^{-x}}{a}{d}{q} }
lim N â â â¡ Q n ⢠( q - x ; b , - b - 1 ⢠c - 1 ⢠q - N - 1 , N | q ) = M n â¡ ( q - x ; b , c ; q ) subscript â ð subscript ð ð superscript ð ð¥ ð superscript ð 1 superscript ð 1 superscript ð ð 1 conditional ð ð q-Meixner-polynomial-M ð superscript ð ð¥ ð ð ð {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}(q^{-x% };b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_{n}\!\left(q^{-x};b,c;q\right)}}} {\displaystyle \lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=\qMeixner{n}@{q^{-x}}{b}{c}{q} }
lim c â â â¡ M n â¡ ( c ⢠q α ⢠x ; q α , c ; q ) = ( q ; q ) n ( q α + 1 ; q ) n ⢠L n ( α ) â¡ ( x ; q ) subscript â ð q-Meixner-polynomial-M ð ð superscript ð ð¼ ð¥ superscript ð ð¼ ð ð q-Pochhammer-symbol ð ð ð q-Pochhammer-symbol superscript ð ð¼ 1 ð ð q-Laguerre-polynomial-L ð¼ ð ð¥ ð {\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow\infty}M_{n}\!% \left(cq^{\alpha}x;q^{\alpha},c;q\right)=\frac{\left(q;q\right)_{n}}{\left(q^{% \alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)}}} {\displaystyle \lim_{c\rightarrow\infty}\qMeixner{n}@{cq^{\alpha}x}{q^{\alpha}}{c}{q}= \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q} }
\qMeixner ⢠n ⢠@ ⢠x ⢠0 ⢠c ⢠q = \qCharlier ⢠n ⢠@ ⢠x ⢠c ⢠q \qMeixner ð @ ð¥ 0 ð ð \qCharlier ð @ ð¥ ð ð {\displaystyle{\displaystyle{\displaystyle\qMeixner{n}@{x}{0}{c}{q}=\qCharlier% {n}@{x}{c}{q}}}} {\displaystyle \qMeixner{n}@{x}{0}{c}{q}=\qCharlier{n}@{x}{c}{q} }
lim c â 0 â¡ M n â¡ ( x ; - a ⢠c - 1 , c ; q ) = ( - 1 a ) n ⢠q \binomial ⢠n ⢠2 ⢠V n ( a ) â¡ ( x ; q ) subscript â ð 0 q-Meixner-polynomial-M ð ð¥ ð superscript ð 1 ð ð superscript 1 ð ð superscript ð \binomial ð 2 q-Al-Salam-Carlitz-II-polynomial-V ð ð ð¥ ð {\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow 0}M_{n}\!\left(x;% -ac^{-1},c;q\right)=\left(-\frac{1}{a}\right)^{n}q^{\binomial{n}{2}}V^{(a)}_{n% }\!\left(x;q\right)}}} {\displaystyle \lim_{c\rightarrow 0}\qMeixner{n}@{x}{-ac^{-1}}{c}{q}= \left(-\frac{1}{a}\right)^nq^{\binomial{n}{2}}\AlSalamCarlitzII{a}{n}@{x}{q} }
lim q â 1 M n â¡ ( q - x ; q β - 1 , ( 1 - c ) - 1 c ; q ) = M n â¡ ( x ; β , c ) fragments subscript â ð 1 superscript Meixner-polynomial-M ð superscript ð ð¥ superscript ð ð½ 1 fragments ( 1 c 1 c ; q ) Meixner-polynomial-M ð ð¥ ð½ ð {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}M_{n}\!\left(q^% {-x};q^{\beta-1},(1-c\right)^{-1}c;q)=M_{n}\!\left(x;\beta,c\right)}}} {\displaystyle \lim_{q\rightarrow 1}\Meixner{n}@{q^{-x}}{q^{\beta-1}}{(1-c}^{-1}c;q)=\Meixner{n}@{x}{\beta}{c} }
\qMeixner ⢠n ⢠@ ⢠q - x ⢠b ⢠c ⢠q = \littleqJacobi ⢠n ⢠@ - c - 1 ⢠q n ⢠b ⢠b - 1 ⢠q - n - x - 1 ⢠q \qMeixner ð @ superscript ð ð¥ ð ð ð \littleqJacobi ð @ superscript ð 1 superscript ð ð ð superscript ð 1 superscript ð ð ð¥ 1 ð {\displaystyle{\displaystyle{\displaystyle\qMeixner{n}@{q^{-x}}{b}{c}{q}=% \littleqJacobi{n}@{-c^{-1}q^{n}}{b}{b^{-1}q^{-n-x-1}}{q}}}} {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\littleqJacobi{n}@{-c^{-1}q^n}{b}{b^{-1}q^{-n-x-1}}{q} } \qtmqKrawtchouk ⢠n ⢠@ ⢠q - x ⢠p ⢠N ⢠q = \qMeixner ⢠n ⢠@ ⢠q - x ⢠q - N - 1 - p - 1 ⢠q \qtmqKrawtchouk ð @ superscript ð ð¥ ð ð ð \qMeixner ð @ superscript ð ð¥ superscript ð ð 1 superscript ð 1 ð {\displaystyle{\displaystyle{\displaystyle\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}% =\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q}}}} {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q} }