The q-shifted factorial

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The q-shifted factorial

lim q → 1 ⁡ 1 - q α 1 - q = α subscript → 𝑞 1 1 superscript 𝑞 𝛼 1 𝑞 𝛼 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\frac{1-% q^{\alpha}}{1-q}=\alpha}}} {\displaystyle \lim\limits_{q\rightarrow 1}\frac{1-q^{\alpha}}{1-q}=\alpha }
[ α ] := 1 - q α 1 - q assign delimited-[] 𝛼 1 superscript 𝑞 𝛼 1 𝑞 {\displaystyle{\displaystyle{\displaystyle[\alpha]:=\frac{1-q^{\alpha}}{1-q}}}} {\displaystyle [\alpha]:=\frac{1-q^{\alpha}}{1-q} }
[ 0 ] = 0 , [ n ] = 1 - q n 1 - q = ∑ k = 0 n - 1 q k formulae-sequence delimited-[] 0 0 delimited-[] 𝑛 1 superscript 𝑞 𝑛 1 𝑞 superscript subscript 𝑘 0 𝑛 1 superscript 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle[0]=0,\quad[n]=\frac{1-q^{n}}{1-q}=% \sum_{k=0}^{n-1}q^{k}}}} {\displaystyle [0]=0,\quad [n]=\frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}


( a ; q ) 0 := 1   and   ( a ; q ) k := ∏ n = 1 k ( 1 - a ⁢ q n - 1 ) , k = 1 , 2 , 3 , … formulae-sequence assign q-Pochhammer-symbol 𝑎 𝑞 0 1 and formulae-sequence assign q-Pochhammer-symbol 𝑎 𝑞 𝑘 superscript subscript product 𝑛 1 𝑘 1 𝑎 superscript 𝑞 𝑛 1 𝑘 1 2 3 … {\displaystyle{\displaystyle{\displaystyle{}\left(a;q\right)_{0}:=1\quad% \textrm{and}\quad\left(a;q\right)_{k}:=\prod_{n=1}^{k}(1-aq^{n-1}),\quad k=1,2% ,3,\ldots}}} {\displaystyle \index{q-Shifted factorial@$q$-Shifted factorial} \qPochhammer{a}{q}{0}:=1\quad\textrm{and}\quad \qPochhammer{a}{q}{k}:=\prod_{n=1}^k(1-aq^{n-1}),\quad k=1,2,3,\ldots }
lim q → 1 ⁡ ( q α ; q ) k ( 1 - q ) k = ( α ) k subscript → 𝑞 1 q-Pochhammer-symbol superscript 𝑞 𝛼 𝑞 𝑘 superscript 1 𝑞 𝑘 Pochhammer-symbol 𝛼 𝑘 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\frac{% \left(q^{\alpha};q\right)_{k}}{(1-q)^{k}}={\left(\alpha\right)_{k}}}}} {\displaystyle \lim\limits_{q\rightarrow 1}\frac{\qPochhammer{q^{\alpha}}{q}{k}}{(1-q)^k}=\pochhammer{\alpha}{k} }
( a ; q ) - k := 1 ∏ n = 1 k ( 1 - a ⁢ q - n ) , a ≠ q , q 2 , q 3 , … , q k formulae-sequence assign q-Pochhammer-symbol 𝑎 𝑞 𝑘 1 superscript subscript product 𝑛 1 𝑘 1 𝑎 superscript 𝑞 𝑛 𝑎 𝑞 superscript 𝑞 2 superscript 𝑞 3 … superscript 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{-k}:=\frac{1}{% \displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^{2},q^{3},\ldots,q^{k% }}}} {\displaystyle \qPochhammer{a}{q}{-k}:=\frac{1}{\displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^2,q^3,\ldots,q^k }

Constraint(s): k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle{\displaystyle k=1,2,3,\ldots}}}


( a ; q ) - n = 1 ( a ⁢ q - n ; q ) n = ( - q ⁢ a - 1 ) n ( q ⁢ a - 1 ; q ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 1 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 superscript 𝑞 superscript 𝑎 1 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝑎 1 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{-n}=\frac{1}{\left% (aq^{-n};q\right)_{n}}=\frac{(-qa^{-1})^{n}}{\left(qa^{-1};q\right)_{n}}q^{% \binomial{n}{2}}}}} {\displaystyle \qPochhammer{a}{q}{-n}=\frac{1}{\qPochhammer{aq^{-n}}{q}{n}}=\frac{(-qa^{-1})^n}{\qPochhammer{qa^{-1}}{q}{n}} q^{\binomial{n}{2}} }

Constraint(s): n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}} &
a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}}


Substitution(s): ( a ⁢ q - n ; q ) n = ( a - 1 ⁢ q ; q ) n ⁢ ( - a ) n ⁢ q - n - \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 𝑛 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n}=\left(a^{% -1}q;q\right)_{n}(-a)^{n}q^{-n-\binomial{n}{2}}}}}


( a ; q - 1 ) n = ( a - 1 ; q ) n ⁢ ( - a ) n ⁢ q - \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 1 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q^{-1}\right)_{n}=\left(a^{-% 1};q\right)_{n}(-a)^{n}q^{-\binomial{n}{2}}}}} {\displaystyle \qPochhammer{a}{q^{-1}}{n}=\qPochhammer{a^{-1}}{q}{n}(-a)^nq^{-\binomial{n}{2}} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}}


( a ; q ) λ = ( a ; q ) ∞ ( a ⁢ q λ ; q ) ∞ q-Pochhammer-symbol 𝑎 𝑞 𝜆 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝜆 𝑞 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\lambda}=\frac{% \left(a;q\right)_{\infty}}{\left(aq^{\lambda};q\right)_{\infty}}}}} {\displaystyle \qPochhammer{a}{q}{\lambda}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^{\lambda}}{q}{\infty}} }

Constraint(s): 0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


( a ; q ) n + k = ( a ; q ) n ⁢ ( a ⁢ q n ; q ) k q-Pochhammer-symbol 𝑎 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n+k}=\left(a;q% \right)_{n}\left(aq^{n};q\right)_{k}}}} {\displaystyle \qPochhammer{a}{q}{n+k}=\qPochhammer{a}{q}{n}\qPochhammer{aq^n}{q}{k} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &
0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) n = ( a ; q ) ∞ ( a ⁢ q n ; q ) ∞ = ( a - 1 ⁢ q 1 - n ; q ) n ⁢ ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &
( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


( a ⁢ q n ; q ) k ( a ⁢ q k ; q ) n = ( a ; q ) k ( a ; q ) n q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑘 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(aq^{n};q\right)_{k}}{% \left(aq^{k};q\right)_{n}}=\frac{\left(a;q\right)_{k}}{\left(a;q\right)_{n}}}}} {\displaystyle \frac{\qPochhammer{aq^n}{q}{k}}{\qPochhammer{aq^k}{q}{n}}=\frac{\qPochhammer{a}{q}{k}}{\qPochhammer{a}{q}{n}} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &
0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) n = ( a ; q ) ∞ ( a ⁢ q n ; q ) ∞ = ( a - 1 ⁢ q 1 - n ; q ) n ⁢ ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &
( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


( a ⁢ q k ; q ) n - k = ( a ; q ) n ( a ; q ) k q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑘 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(aq^{k};q\right)_{n-k}=\frac{% \left(a;q\right)_{n}}{\left(a;q\right)_{k}}}}} {\displaystyle \qPochhammer{aq^k}{q}{n-k}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{a}{q}{k}} }

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

0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}} &
a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &

0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) n = ( a ; q ) ∞ ( a ⁢ q n ; q ) ∞ = ( a - 1 ⁢ q 1 - n ; q ) n ⁢ ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &
( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


( a ⁢ q - n ; q ) n ( b ⁢ q - n ; q ) n = ( a - 1 ⁢ q ; q ) n ( b - 1 ⁢ q ; q ) n ⁢ ( a b ) n , a ≠ 0 formulae-sequence q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol 𝑏 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑏 1 𝑞 𝑞 𝑛 superscript 𝑎 𝑏 𝑛 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\frac{\left(aq^{-n};q\right)_{n}}{% \left(bq^{-n};q\right)_{n}}=\frac{\left(a^{-1}q;q\right)_{n}}{\left(b^{-1}q;q% \right)_{n}}\left(\frac{a}{b}\right)^{n},\quad a\neq 0}}} {\displaystyle \frac{\qPochhammer{aq^{-n}}{q}{n}}{\qPochhammer{bq^{-n}}{q}{n}}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{b^{-1}q}{q}{n}} \left(\frac{a}{b}\right)^n,\quad a\neq 0 }

Constraint(s): b ≠ 0 𝑏 0 {\displaystyle{\displaystyle{\displaystyle b\neq 0}}} &
a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}}


Substitution(s): ( a ⁢ q - n ; q ) n = ( a - 1 ⁢ q ; q ) n ⁢ ( - a ) n ⁢ q - n - \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 𝑛 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n}=\left(a^{% -1}q;q\right)_{n}(-a)^{n}q^{-n-\binomial{n}{2}}}}}


( a ; q ) n - k ( b ; q ) n - k = ( a ; q ) n ( b ; q ) n ⁢ ( b - 1 ⁢ q 1 - n ; q ) k ( a - 1 ⁢ q 1 - n ; q ) k ⁢ ( b a ) k q-Pochhammer-symbol 𝑎 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑏 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑏 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑏 1 superscript 𝑞 1 𝑛 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑘 superscript 𝑏 𝑎 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{\left(a;q\right)_{n-k}}{\left(% b;q\right)_{n-k}}=\frac{\left(a;q\right)_{n}}{\left(b;q\right)_{n}}\frac{\left% (b^{-1}q^{1-n};q\right)_{k}}{\left(a^{-1}q^{1-n};q\right)_{k}}\left(\frac{b}{a% }\right)^{k}{}}}} {\displaystyle \frac{\qPochhammer{a}{q}{n-k}}{\qPochhammer{b}{q}{n-k}}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{b}{q}{n}} \frac{\qPochhammer{b^{-1}q^{1-n}}{q}{k}}{\qPochhammer{a^{-1}q^{1-n}}{q}{k}}\left(\frac{b}{a}\right)^k {} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &

b ≠ 0 𝑏 0 {\displaystyle{\displaystyle{\displaystyle b\neq 0}}} &
k = 0 , 1 , 2 , … , n 𝑘 0 1 2 … 𝑛 {\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}} &
0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}} &
a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &

0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) n - k = ( a ; q ) n ( a - 1 ⁢ q 1 - n ; q ) k ⁢ ( - q a ) k ⁢ q \binomial ⁢ k ⁢ 2 - n ⁢ k , a ≠ 0 formulae-sequence q-Pochhammer-symbol 𝑎 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑘 superscript 𝑞 𝑎 𝑘 superscript 𝑞 \binomial 𝑘 2 𝑛 𝑘 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n-k}=\frac{\left(a% ;q\right)_{n}}{\left(a^{-1}q^{1-n};q\right)_{k}}\left(-\frac{q}{a}\right)^{k}q% ^{\binomial{k}{2}-nk},\quad a\neq 0}}} &

( a ; q ) n = ( a ; q ) ∞ ( a ⁢ q n ; q ) ∞ = ( a - 1 ⁢ q 1 - n ; q ) n ⁢ ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &

( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


( q - n ; q ) k = ( q ; q ) n ( q ; q ) n - k ⁢ ( - 1 ) k ⁢ q \binomial ⁢ k ⁢ 2 - n ⁢ k q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 superscript 1 𝑘 superscript 𝑞 \binomial 𝑘 2 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(q^{-n};q\right)_{k}=\frac{% \left(q;q\right)_{n}}{\left(q;q\right)_{n-k}}(-1)^{k}q^{\binomial{k}{2}-nk}}}} {\displaystyle \qPochhammer{q^{-n}}{q}{k}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{n-k}}(-1)^kq^{\binomial{k}{2}-nk} }

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


( a ⁢ q - n ; q ) k = ( a - 1 ⁢ q ; q ) n ( a - 1 ⁢ q 1 - k ; q ) n ⁢ ( a ; q ) k ⁢ q - n ⁢ k q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑘 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑘 superscript 𝑞 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{k}=\frac{% \left(a^{-1}q;q\right)_{n}}{\left(a^{-1}q^{1-k};q\right)_{n}}\left(a;q\right)_% {k}q^{-nk}}}} {\displaystyle \qPochhammer{aq^{-n}}{q}{k}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{a^{-1}q^{1-k}}{q}{n}}\qPochhammer{a}{q}{k}q^{-nk} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}}


( a ⁢ q - n ; q ) n - k = ( a - 1 ⁢ q ; q ) n ( a - 1 ⁢ q ; q ) k ⁢ ( - a q ) n - k ⁢ q \binomial ⁢ k ⁢ 2 - \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 𝑘 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑘 superscript 𝑎 𝑞 𝑛 𝑘 superscript 𝑞 \binomial 𝑘 2 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n-k}=\frac{% \left(a^{-1}q;q\right)_{n}}{\left(a^{-1}q;q\right)_{k}}\left(-\frac{a}{q}% \right)^{n-k}q^{\binomial{k}{2}-\binomial{n}{2}}{}}}} {\displaystyle \qPochhammer{aq^{-n}}{q}{n-k}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{a^{-1}q}{q}{k}} \left(-\frac{a}{q}\right)^{n-k}q^{\binomial{k}{2}-\binomial{n}{2}} {} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &
k = 0 , 1 , 2 , … , n 𝑘 0 1 2 … 𝑛 {\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}}


( a ; q ) 2 ⁢ n = ( a ; q 2 ) n ⁢ ( a ⁢ q ; q 2 ) n q-Pochhammer-symbol 𝑎 𝑞 2 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 𝑛 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{2n}=\left(a;q^{2}% \right)_{n}\left(aq;q^{2}\right)_{n}}}} {\displaystyle \qPochhammer{a}{q}{2n}=\qPochhammer{a}{q^2}{n}\qPochhammer{aq}{q^2}{n} }
( a 2 ; q 2 ) ∞ = ( a ; q ) ∞ ⁢ ( - a ; q ) ∞ q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle\left(a^{2};q^{2}\right)_{\infty}=% \left(a;q\right)_{\infty}\left(-a;q\right)_{\infty}}}} {\displaystyle \qPochhammer{a^2}{q^2}{\infty}=\qPochhammer{a}{q}{\infty}\qPochhammer{-a}{q}{\infty} }

Constraint(s): 0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


1 - a 2 ⁢ q 2 ⁢ n 1 - a 2 = ( a 2 ⁢ q 2 ; q 2 ) n ( a 2 ; q 2 ) n = ( a ⁢ q ; q ) n ⁢ ( - a ⁢ q ; q ) n ( a ; q ) n ⁢ ( - a ; q ) n 1 superscript 𝑎 2 superscript 𝑞 2 𝑛 1 superscript 𝑎 2 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 2 superscript 𝑞 2 𝑛 q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 2 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1-a^{2}q^{2n}}{1-a^{2}}=\frac{% \left(a^{2}q^{2};q^{2}\right)_{n}}{\left(a^{2};q^{2}\right)_{n}}=\frac{\left(% aq;q\right)_{n}\left(-aq;q\right)_{n}}{\left(a;q\right)_{n}\left(-a;q\right)_{% n}}}}} {\displaystyle \frac{1-a^2q^{2n}}{1-a^2}=\frac{\qPochhammer{a^2q^2}{q^2}{n}}{\qPochhammer{a^2}{q^2}{n}} =\frac{\qPochhammer{aq}{q}{n}\qPochhammer{-aq}{q}{n}}{\qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}} }

Constraint(s): a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &
0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s): ( a 2 ; q 2 ) n = ( a ; q ) n ⁢ ( - a ; q ) n q-Pochhammer-symbol superscript 𝑎 2 superscript 𝑞 2 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(a^{2};q^{2}\right)_{n}=\left(a% ;q\right)_{n}\left(-a;q\right)_{n}}}} &

( a ; q ) n = ( a ; q ) ∞ ( a ⁢ q n ; q ) ∞ = ( a - 1 ⁢ q 1 - n ; q ) n ⁢ ( - a ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &

( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a ⁢ q k ) = ( a ; q 2 ) ∞ ⁢ ( a ⁢ q ; q 2 ) ∞ q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


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