q-Hahn

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q-Hahn

Basic hypergeometric representation

Q n ⁡ ( q - x ; α , β , N ; q ) = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , α ⁢ β ⁢ q n + 1 , q - x ⁢ α ⁢ q , q - N ⁢ q ⁢ q q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 superscript 𝑞 𝑥 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right)=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-% N}}{q}{q}}}} {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-N}}{q}{q} }

Constraint(s): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


Orthogonality relation(s)

∑ x = 0 N ( α ⁢ q , q - N ; q ) x ( q , β - 1 ⁢ q - N ; q ) x ⁢ ( α ⁢ β ⁢ q ) - x ⁢ Q m ⁡ ( q - x ; α , β , N ; q ) ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) = ( α ⁢ β ⁢ q 2 ; q ) N ( β ⁢ q ; q ) N ⁢ ( α ⁢ q ) N ⁢ ( q , α ⁢ β ⁢ q N + 2 , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , q - N ; q ) n ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( - α ⁢ q ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ q \binomial ⁢ n ⁢ 2 - N ⁢ n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛽 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝛼 𝛽 𝑞 𝑥 q-Hahn-polynomial-Q 𝑚 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 2 𝑞 𝑁 q-Pochhammer-symbol 𝛽 𝑞 𝑞 𝑁 superscript 𝛼 𝑞 𝑁 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝑞 𝑁 2 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 1 𝛼 𝛽 𝑞 superscript 𝛼 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 superscript 𝑞 \binomial 𝑛 2 𝑁 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\alpha q,q% ^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha\beta q)^{-x}% Q_{m}\!\left(q^{-x};\alpha,\beta,N;q\right)Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right){}=\frac{\left(\alpha\beta q^{2};q\right)_{N}}{\left(\beta q;q\right)_% {N}(\alpha q)^{N}}\frac{\left(q,\alpha\beta q^{N+2},\beta q;q\right)_{n}}{% \left(\alpha q,\alpha\beta q,q^{-N};q\right)_{n}}\ \frac{(1-\alpha\beta q)(-% \alpha q)^{n}}{(1-\alpha\beta q^{2n+1})}q^{\binomial{n}{2}-Nn}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N}}{q}{x}}(\alpha\beta q)^{-x} \qHahn{m}@{q^{-x}}{\alpha}{\beta}{N}{q}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=\frac{\qPochhammer{\alpha\beta q^2}{q}{N}}{\qPochhammer{\beta q}{q}{N}(\alpha q)^N} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,q^{-N}}{q}{n}}\ \frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} q^{\binomial{n}{2}-Nn}\,\Kronecker{m}{n} }

Recurrence relation

- ( 1 - q - x ) ⁢ Q n ⁡ ( q - x ) = A n ⁢ Q n + 1 ⁡ ( q - x ) - ( A n + C n ) ⁢ Q n ⁡ ( q - x ) + C n ⁢ Q n - 1 ⁡ ( q - x ) 1 superscript 𝑞 𝑥 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐶 𝑛 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)Q_{n}\!\left(q% ^{-x}\right)=A_{n}Q_{n+1}\!\left(q^{-x}\right)-\left(A_{n}+C_{n}\right)Q_{n}\!% \left(q^{-x}\right){}+C_{n}Q_{n-1}\!\left(q^{-x}\right)}}} {\displaystyle -\left(1-q^{-x}\right)\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q}=A_n\qHahn{n+1}@@{q^{-x}}{\alpha}{\beta}{N}{q}-\left(A_n+C_n\right)\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q} {}+C_n\qHahn{n-1}@@{q^{-x}}{\alpha}{\beta}{N}{q} }

Substitution(s): C n = - α ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + N + 1 ) ⁢ ( 1 - β ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝛼 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 𝑁 1 1 𝛽 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-\frac{\alpha q^{n-N}(1-q^{n}% )(1-\alpha\beta q^{n+N+1})(1-\beta q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha% \beta q^{2n+1})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1-\alpha q^% {n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2}% )}}}}


Q n ⁡ ( q - x ) := Q n ⁡ ( q - x ; α , β , N ; q ) assign q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x}\right):=Q_{n}\!% \left(q^{-x};\alpha,\beta,N;q\right)}}} {\displaystyle \qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q}:=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }

Monic recurrence relation

x ⁢ Q ^ n ⁡ ( x ) ⁢ q = Q ^ n + 1 ⁡ ( x ) ⁢ q + [ 1 - ( A n + C n ) ] ⁢ Q ^ n ⁡ ( x ) ⁢ q + A n - 1 ⁢ C n ⁢ Q ^ n - 1 ⁡ ( x ) ⁢ q 𝑥 q-Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 𝑞 q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right){% q}={\widehat{Q}}_{n+1}\!\left(x\right){q}+\left[1-(A_{n}+C_{n})\right]{% \widehat{Q}}_{n}\!\left(x\right){q}+A_{n-1}C_{n}{\widehat{Q}}_{n-1}\!\left(x% \right){q}}}} {\displaystyle x\monicqHahn{n}@@{x}{\alpha}{\beta}{N}{q}=\monicqHahn{n+1}@@{x}{\alpha}{\beta}{N}{q}+\left[1-(A_n+C_n)\right]\monicqHahn{n}@@{x}{\alpha}{\beta}{N}{q}+A_{n-1}C_n\monicqHahn{n-1}@@{x}{\alpha}{\beta}{N}{q} }

Substitution(s): C n = - α ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + N + 1 ) ⁢ ( 1 - β ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝛼 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 𝑁 1 1 𝛽 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-\frac{\alpha q^{n-N}(1-q^{n}% )(1-\alpha\beta q^{n+N+1})(1-\beta q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha% \beta q^{2n+1})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1-\alpha q^% {n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2}% )}}}}


Q n ⁡ ( q - x ; α , β , N ; q ) = ( α ⁢ β ⁢ q n + 1 ; q ) n ( α ⁢ q , q - N ; q ) n ⁢ Q ^ n ⁡ ( q - x ) ⁢ q q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 q-Hahn-polynomial-monic-p 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right)=\frac{\left(\alpha\beta q^{n+1};q\right)_{n}}{\left(\alpha q,q^{-N};q% \right)_{n}}{\widehat{Q}}_{n}\!\left(q^{-x}\right){q}}}} {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}= \frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,q^{-N}}{q}{n}}\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q} }

q-Difference equation

q - n ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + 1 ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - 1 ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 1 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-\alpha\beta q^{n+% 1})y(x){}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}} {\displaystyle q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})y(x) {}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1) }

Substitution(s): D ⁢ ( x ) = α ⁢ q ⁢ ( 1 - q x ) ⁢ ( β - q x - N - 1 ) 𝐷 𝑥 𝛼 𝑞 1 superscript 𝑞 𝑥 𝛽 superscript 𝑞 𝑥 𝑁 1 {\displaystyle{\displaystyle{\displaystyle D(x)=\alpha q(1-q^{x})(\beta-q^{x-N% -1})}}} &

B ⁢ ( x ) = ( 1 - q x - N ) ⁢ ( 1 - α ⁢ q x + 1 ) 𝐵 𝑥 1 superscript 𝑞 𝑥 𝑁 1 𝛼 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})}}} &

y ⁢ ( x ) = Q n ⁡ ( q - x ; α , β , N ; q ) 𝑦 𝑥 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=Q_{n}\!\left(q^{-x};\alpha,% \beta,N;q\right)}}}


Forward shift operator

Q n ⁡ ( q - x - 1 ; α , β , N ; q ) - Q n ⁡ ( q - x ; α , β , N ; q ) = q - n - x ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - α ⁢ q ) ⁢ ( 1 - q - N ) ⁢ Q n - 1 ⁡ ( q - x ; α ⁢ q , β ⁢ q , N - 1 ; q ) q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 1 𝛼 𝛽 𝑁 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛼 𝑞 1 superscript 𝑞 𝑁 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝑞 𝛽 𝑞 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x-1};\alpha,\beta,% N;q\right)-Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right){}=\frac{q^{-n-x}(1-q^{n% })(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})}Q_{n-1}\!\left(q^{-x};\alpha q% ,\beta q,N-1;q\right)}}} {\displaystyle \qHahn{n}@{q^{-x-1}}{\alpha}{\beta}{N}{q}-\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} \qHahn{n-1}@{q^{-x}}{\alpha q}{\beta q}{N-1}{q} }
Δ ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) Δ ⁢ q - x = q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - q ) ⁢ ( 1 - α ⁢ q ) ⁢ ( 1 - q - N ) ⁢ Q n - 1 ⁡ ( q - x ; α ⁢ q , β ⁢ q , N - 1 ; q ) Δ q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 Δ superscript 𝑞 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝑞 1 𝛼 𝑞 1 superscript 𝑞 𝑁 q-Hahn-polynomial-Q 𝑛 1 superscript 𝑞 𝑥 𝛼 𝑞 𝛽 𝑞 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta Q_{n}\!\left(q^{-x};% \alpha,\beta,N;q\right)}{\Delta q^{-x}}=\frac{q^{-n+1}(1-q^{n})(1-\alpha\beta q% ^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})}{}Q_{n-1}\!\left(q^{-x};\alpha q,\beta q,% N-1;q\right)}}} {\displaystyle \frac{\Delta \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}}{\Delta q^{-x}} =\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})} {} \qHahn{n-1}@{q^{-x}}{\alpha q}{\beta q}{N-1}{q} }

Backward shift operator

( 1 - α ⁢ q x ) ⁢ ( 1 - q x - N - 1 ) ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) - α ⁢ ( 1 - q x ) ⁢ ( β - q x - N - 1 ) ⁢ Q n ⁡ ( q - x + 1 ; α , β , N ; q ) = q x ⁢ ( 1 - α ) ⁢ ( 1 - q - N - 1 ) ⁢ Q n + 1 ⁢ ( q - x ; α ⁢ q - 1 , β ⁢ q - 1 , N + 1 | q ) 1 𝛼 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 𝑁 1 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 𝛼 1 superscript 𝑞 𝑥 𝛽 superscript 𝑞 𝑥 𝑁 1 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 1 𝛼 𝛽 𝑁 𝑞 superscript 𝑞 𝑥 1 𝛼 1 superscript 𝑞 𝑁 1 subscript 𝑄 𝑛 1 superscript 𝑞 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝑁 conditional 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-\alpha q^{x})(1-q^{x-N-1})Q_{n}\!% \left(q^{-x};\alpha,\beta,N;q\right){}-\alpha(1-q^{x})(\beta-q^{x-N-1})Q_{n}\!% \left(q^{-x+1};\alpha,\beta,N;q\right){}=q^{x}(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^% {-x};\alpha q^{-1},\beta q^{-1},N+1|q)}}} {\displaystyle (1-\alpha q^x)(1-q^{x-N-1})\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}-\alpha(1-q^x)(\beta-q^{x-N-1})\qHahn{n}@{q^{-x+1}}{\alpha}{\beta}{N}{q} {}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) }
∇ ⁡ [ w ⁢ ( x ; α , β , N | q ) ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) ] ∇ ⁡ q - x = 1 1 - q ⁢ w ⁢ ( x ; α ⁢ q - 1 , β ⁢ q - 1 , N + 1 | q ) ⁢ Q n + 1 ⁢ ( q - x ; α ⁢ q - 1 , β ⁢ q - 1 , N + 1 | q ) ∇ 𝑤 𝑥 𝛼 𝛽 conditional 𝑁 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 ∇ superscript 𝑞 𝑥 1 1 𝑞 𝑤 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝑁 conditional 1 𝑞 subscript 𝑄 𝑛 1 superscript 𝑞 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝑁 conditional 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;\alpha,\beta,N% |q)Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)\right]}{\nabla q^{-x}}{}=\frac{% 1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q% ^{-1},N+1|q)}}} {\displaystyle \frac{\nabla\left[w(x;\alpha,\beta,N|q)\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}\right]}{\nabla q^{-x}} {}=\frac{1}{1-q}w(x;\alpha q^{-1},\beta q^{-1},N+1|q)Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) }

Substitution(s): w ⁢ ( x ; α , β , N | q ) = ( α ⁢ q , q - N ; q ) x ( q , β - 1 ⁢ q - N ; q ) x ⁢ ( α ⁢ β ) - x 𝑤 𝑥 𝛼 𝛽 conditional 𝑁 𝑞 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛽 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝛼 𝛽 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\left(% \alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha% \beta)^{-x}}}}


Rodrigues-type formula

w ⁢ ( x ; α , β , N | q ) ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) = ( 1 - q ) n ⁢ ( ∇ q ) n ⁢ [ w ⁢ ( x ; α ⁢ q n , β ⁢ q n , N - n | q ) ] 𝑤 𝑥 𝛼 𝛽 conditional 𝑁 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript 1 𝑞 𝑛 superscript subscript ∇ 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝛼 superscript 𝑞 𝑛 𝛽 superscript 𝑞 𝑛 𝑁 conditional 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)Q_{n}\!\left(q% ^{-x};\alpha,\beta,N;q\right){}=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;% \alpha q^{n},\beta q^{n},N-n|q)\right]}}} {\displaystyle w(x;\alpha,\beta,N|q)\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right] }

Substitution(s): w ⁢ ( x ; α , β , N | q ) = ( α ⁢ q , q - N ; q ) x ( q , β - 1 ⁢ q - N ; q ) x ⁢ ( α ⁢ β ) - x 𝑤 𝑥 𝛼 𝛽 conditional 𝑁 𝑞 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛽 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝛼 𝛽 𝑥 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\left(% \alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha% \beta)^{-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 ⁢ 11 ⁢ @ ⁢ @ ⁢ q - x ⁢ α ⁢ q ⁢ q ⁢ α ⁢ q ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q x - N , 0 ⁢ β ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( β ⁢ q , q ; q ) n ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) ⁢ t n \qHyperrphis 11 @ @ superscript 𝑞 𝑥 𝛼 𝑞 𝑞 𝛼 𝑞 𝑡 \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝑁 0 𝛽 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝛽 𝑞 𝑞 𝑞 𝑛 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{1}{1}@@{q^{-x}}{\alpha q% }{q}{\alpha qt}\,\qHyperrphis{2}{1}@@{q^{x-N},0}{\beta q}{q}{q^{-x}t}{}=\sum_{% n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{\left(\beta q,q;q\right)_{n}}Q_{n}\!% \left(q^{-x};\alpha,\beta,N;q\right)t^{n}}}} {\displaystyle \qHyperrphis{1}{1}@@{q^{-x}}{\alpha q}{q}{\alpha qt}\,\qHyperrphis{2}{1}@@{q^{x-N},0}{\beta q}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{\beta q,q}{q}{n}}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}t^n }
\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , β ⁢ q N + 1 - x ⁢ 0 ⁢ q - α ⁢ q x - N + 1 ⁢ t ⁢ \qHyperrphis ⁢ 20 ⁢ @ ⁢ @ ⁢ q x - N , α ⁢ q x + 1 - q - q - x ⁢ t = ∑ n = 0 N ( α ⁢ q , q - N ; q ) n ( q ; q ) n ⁢ q - \binomial ⁢ n ⁢ 2 ⁢ Q n ⁡ ( q - x ; α , β , N ; q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝛽 superscript 𝑞 𝑁 1 𝑥 0 𝑞 𝛼 superscript 𝑞 𝑥 𝑁 1 𝑡 \qHyperrphis 20 @ @ superscript 𝑞 𝑥 𝑁 𝛼 superscript 𝑞 𝑥 1 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\beta q^% {N+1-x}}{0}{q}{-\alpha q^{x-N+1}t}\ \qHyperrphis{2}{0}@@{q^{x-N},\alpha q^{x+1% }}{-}{q}{-q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(\alpha q,q^{-N};q\right)_{n}}{% \left(q;q\right)_{n}}q^{-\binomial{n}{2}}Q_{n}\!\left(q^{-x};\alpha,\beta,N;q% \right)t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{q^{-x},\beta q^{N+1-x}}{0}{q}{-\alpha q^{x-N+1}t}\ \qHyperrphis{2}{0}@@{q^{x-N},\alpha q^{x+1}}{-}{q}{-q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{\alpha q,q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{-\binomial{n}{2}}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}t^n }

Limit relations

q-Racah polynomial to q-Hahn polynomial

R n ⁡ ( μ ⁢ ( x ) ; α , β , q - N - 1 , 0 | q ) = Q n ⁡ ( q - x ; α , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 superscript 𝑞 𝑁 1 0 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,q^% {-N-1},0\,|\,q\right)=Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)}}} {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{q^{-N-1}}{0}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( n ) = q - n + α ⁢ β ⁢ q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}}}}


R n ⁡ ( μ ⁢ ( x ) ; α , β , 0 , β - 1 ⁢ q - N - 1 | q ) = Q n ⁡ ( q - x ; α , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 0 superscript 𝛽 1 superscript 𝑞 𝑁 1 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,0,% \beta^{-1}q^{-N-1}\,|\,q\right)=Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)}}} {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{0}{\beta^{-1}q^{-N-1}}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( n ) = q - n + α ⁢ β ⁢ q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}}}}


R n ⁡ ( μ ⁢ ( x ) ; q - N - 1 , β ⁢ γ ⁢ q N + 1 , γ , 0 | q ) = Q n ⁡ ( q - x ; γ , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 superscript 𝑞 𝑁 1 𝛽 𝛾 superscript 𝑞 𝑁 1 𝛾 0 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛾 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);q^{-N-1},\beta% \gamma q^{N+1},\gamma,0\,|\,q\right)=Q_{n}\!\left(q^{-x};\gamma,\beta,N;q% \right)}}} {\displaystyle \qRacah{n}@{\mu(x)}{q^{-N-1}}{\beta\gamma q^{N+1}}{\gamma}{0}{q}=\qHahn{n}@{q^{-x}}{\gamma}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( n ) = q - n + α ⁢ β ⁢ q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}}}}


q-Hahn polynomial to Little q-Jacobi polynomial

lim N → ∞ ⁡ Q n ⁡ ( q x - N ; α , β , N ; q ) = p n ⁡ ( q x ; α , β ; q ) subscript → 𝑁 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝑁 𝛼 𝛽 𝑁 𝑞 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(q^{x-N};\alpha,\beta,N;q\right)=p_{n}\!\left(q^{x};\alpha,\beta;q\right)% }}} {\displaystyle \lim _{N\rightarrow\infty} \qHahn{n}@{q^{x-N}}{\alpha}{\beta}{N}{q}=\littleqJacobi{n}@{q^x}{\alpha}{\beta}{q} }

q-Hahn polynomial to q-Meixner polynomial

lim N → ∞ ⁡ Q n ⁢ ( q - x ; b , - b - 1 ⁢ c - 1 ⁢ q - N - 1 , N | q ) = M n ⁡ ( q - x ; b , c ; q ) subscript → 𝑁 subscript 𝑄 𝑛 superscript 𝑞 𝑥 𝑏 superscript 𝑏 1 superscript 𝑐 1 superscript 𝑞 𝑁 1 conditional 𝑁 𝑞 q-Meixner-polynomial-M 𝑛 superscript 𝑞 𝑥 𝑏 𝑐 𝑞 {\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} }

q-Hahn polynomial to Quantum q-Krawtchouk polynomial

lim α → ∞ ⁡ Q n ⁡ ( q - x ; α , p , N ; q ) = K n qtm ⁡ ( q - x ; p , N ; q ) subscript → 𝛼 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}Q_{n}% \!\left(q^{-x};\alpha,p,N;q\right)=K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q% \right)}}} {\displaystyle \lim_{\alpha\rightarrow\infty}\qHahn{n}@{q^{-x}}{\alpha}{p}{N}{q}=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

q-Hahn polynomial to q-Krawtchouk polynomial

lim α → 0 ⁡ Q n ⁢ ( q - x ; α , - α - 1 ⁢ q - 1 ⁢ p , N | q ) = K n ⁡ ( q - x ; p , N ; q ) subscript → 𝛼 0 subscript 𝑄 𝑛 superscript 𝑞 𝑥 𝛼 superscript 𝛼 1 superscript 𝑞 1 𝑝 conditional 𝑁 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow 0}Q_{n}(q^{-% x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \lim_{\alpha\rightarrow 0} Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

q-Hahn polynomial to Affine q-Krawtchouk polynomial

Q n ⁡ ( q - x ; p , 0 , N ; q ) = K n Aff ⁡ ( q - x ; p , N ; q ) q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝑝 0 𝑁 𝑞 affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};p,0,N;q\right)=% K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}} {\displaystyle \qHahn{n}@{q^{-x}}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }

q-Hahn polynomial to Hahn polynomial

lim q → 1 ⁡ Q n ⁢ ( q - x ; q α , q β , N | q ) = Q n ⁡ ( x ; α , β , N ) subscript → 𝑞 1 subscript 𝑄 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 conditional 𝑁 𝑞 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}Q_{n}(q^{-x};q^% {\alpha},q^{\beta},N|q)=Q_{n}\!\left(x;\alpha,\beta,N\right)}}} {\displaystyle \lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=\Hahn{n}@{x}{\alpha}{\beta}{N} }

Remark

Q n ⁡ ( q - x ; α , β , N ; q ) = R x ⁡ ( μ ⁢ ( n ) ; α , β , N ) ⁢ q q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 dual-q-Hahn-R 𝑥 𝜇 𝑛 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;% q\right)=R_{x}\!\left(\mu(n);\alpha,\beta,N\right){q}}}} {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=\dualqHahn{x}@{\mu(n)}{\alpha}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( n ) = q - n + α ⁢ β ⁢ q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}}}}


R n ⁡ ( μ ⁢ ( x ) ; γ , δ , N ) ⁢ q = Q x ⁡ ( q - n ; γ , δ , N ; q ) dual-q-Hahn-R 𝑛 𝜇 𝑥 𝛾 𝛿 𝑁 𝑞 q-Hahn-polynomial-Q 𝑥 superscript 𝑞 𝑛 𝛾 𝛿 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\gamma,\delta,N% \right){q}=Q_{x}\!\left(q^{-n};\gamma,\delta,N;q\right)}}} {\displaystyle \dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}=\qHahn{x}@{q^{-n}}{\gamma}{\delta}{N}{q} }

Substitution(s): μ ⁢ ( n ) = q - n + α ⁢ β ⁢ q n + 1 𝜇 𝑛 superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}}}}


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