§17.2 Calculus

For $n=0,1,2,\mathrm{\dots }$,

17.2.1		$${(a;q)}_n=(1-a)(1-aq)\mathrm{\cdots }(1-a{q}^{n-1}),$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E1 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.2		$${(a;q)}_{-n}=\frac{1}{{(a{q}^{-n};q)}_n}=\frac{{(-q/a)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}}{{(q/a;q)}_n}.$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E2 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

For $\nu \in ℂ$

17.2.3		$${(a;q)}_{\nu }=\prod _{j=0}^{\mathrm{\infty }}\left(\frac{1-a{q}^j}{1-a{q}^{\nu +j}}\right),$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $j$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E3 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

when this product converges.

17.2.4		${(a;q)}_{\mathrm{\infty }}$	$={\displaystyle \prod _{j=0}^{\mathrm{\infty }}}(1-a{q}^j),$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $j$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E4 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17
17.2.5		${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$	$={\displaystyle \prod _{j=1}^r}{(a_j;q)}_n,$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $r$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E5 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17
17.2.6		${(a_1,a_2,\mathrm{\dots },a_r;q)}_{\mathrm{\infty }}$	$={\displaystyle \prod _{j=1}^r}{(a_j;q)}_{\mathrm{\infty }}.$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer and $r$: nonnegative integer Referenced by: §17.2(i) Permalink: http://dlmf.nist.gov/17.2.E6 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17
For properties of the function $f\left(q\right)={(q;q)}_{\mathrm{\infty }}$ see §27.14(ii).

17.2.7		$${(a;q^{-1})}_n={(a^{-1};q)}_n{(-a)}^n{q}^{-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E7 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.8		$$\frac{{(a;q^{-1})}_n}{{(b;q^{-1})}_n}=\frac{{(a^{-1};q)}_n}{{(b^{-1};q)}_n}{\left(\frac{a}{b}\right)}^n,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E8 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.9		$${(a;q)}_n={(q^{1-n}/a;q)}_n{(-a)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E9 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.10		$$\frac{{(a;q)}_n}{{(b;q)}_n}=\frac{{(q^{1-n}/a;q)}_n}{{(q^{1-n}/b;q)}_n}{\left(\frac{a}{b}\right)}^n,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E10 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.11		$${(a{q}^{-n};q)}_n={(q/a;q)}_n{\left(-\frac{a}{q}\right)}^n{q}^{-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E11 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.12		$$\frac{{(a{q}^{-n};q)}_n}{{(b{q}^{-n};q)}_n}=\frac{{(q/a;q)}_n}{{(q/b;q)}_n}{\left(\frac{a}{b}\right)}^n.$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E12 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.13		$${(a;q)}_{n-k}=\frac{{(a;q)}_n}{{(q^{1-n}/a;q)}_k}{\left(-\frac{q}{a}\right)}^k{q}^{\left(\genfrac{}{}{0.0pt}{}{k}{2}\right)-nk},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E13 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.14		$$\frac{{(a;q)}_{n-k}}{{(b;q)}_{n-k}}=\frac{{(a;q)}_n}{{(b;q)}_n}\frac{{(q^{1-n}/b;q)}_k}{{(q^{1-n}/a;q)}_k}{\left(\frac{b}{a}\right)}^k,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E14 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.15		$${(a{q}^{-n};q)}_k=\frac{{(a;q)}_k{(q/a;q)}_n}{{(q^{1-k}/a;q)}_n}{q}^{-nk},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E15 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.16		$${(a{q}^{-n};q)}_{n-k}=\frac{{(q/a;q)}_n}{{(q/a;q)}_k}{\left(-\frac{a}{q}\right)}^{n-k}{q}^{\left(\genfrac{}{}{0.0pt}{}{k}{2}\right)-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E16 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.17		${(a{q}^n;q)}_k$	$={\displaystyle \frac{{(a;q)}_k{(a{q}^k;q)}_n}{{(a;q)}_n}},$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E17 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17
17.2.18		${(a{q}^k;q)}_{n-k}$	$={\displaystyle \frac{{(a;q)}_n}{{(a;q)}_k}}.$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E18 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.19		$${(a;q)}_{2n}={(a,aq;q^2)}_n,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E19 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

more generally,

17.2.20		$${(a;q)}_{kn}={(a,aq,\mathrm{\dots },a{q}^{k-1};q^k)}_n.$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E20 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.21		$${(a^2;q^2)}_n={(a;q)}_n{(-a;q)}_n,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E21 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.22		$$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer Referenced by: Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23) Permalink: http://dlmf.nist.gov/17.2.E22 Encodings: TeX, pMML, png Errata (effective with 1.0.16): The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$. Reported 2017-06-26 by Jason Zhao See also: Annotations for §17.2(i), §17.2 and Ch.17

more generally,

17.2.23		$$\frac{{(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}{{(a^{\frac{1}{k}},{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}=\frac{{(a{q}^k;q^k)}_n}{{(a;q^k)}_n}=\frac{1-a{q}^{kn}}{1-a},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer Referenced by: Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23) Permalink: http://dlmf.nist.gov/17.2.E23 Encodings: TeX, pMML, png Errata (effective with 1.0.16): The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$ instead of ${(a{q}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$. Reported 2017-06-26 by Jason Zhao See also: Annotations for §17.2(i), §17.2 and Ch.17

where ${\omega }_k={\mathrm{e}}^{2\pi \mathrm{i}/k}$.

17.2.24		$$\underset{\tau \to 0}{lim}{(a/\tau ;q)}_n{\tau }^n=\underset{\sigma \to \mathrm{\infty }}{lim}{(a\sigma ;q)}_n{\sigma }^{-n}={(-a)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E24 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.25		$$\underset{\tau \to 0}{lim}\frac{{(a/\tau ;q)}_n}{{(b/\tau ;q)}_n}=\underset{\sigma \to \mathrm{\infty }}{lim}\frac{{(a\sigma ;q)}_n}{{(b\sigma ;q)}_n}={\left(\frac{a}{b}\right)}^n,$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E25 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.26		$$\underset{\tau \to 0}{lim}\frac{{(a/\tau ;q)}_n{(b/\tau ;q)}_n}{{(c/{\tau }^2;q)}_n}={(-1)}^n{\left(\frac{ab}{c}\right)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}.$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E26 Encodings: TeX, pMML, png See also: Annotations for §17.2(i), §17.2 and Ch.17

17.2.27		$${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\frac{{(q;q)}_n}{{(q;q)}_m{(q;q)}_{n-m}}=\frac{{(q^{-n};q)}_m{(-1)}^m{q}^{nm-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}}{{(q;q)}_m},$$
ⓘ Defines: ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial) Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E27 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

17.2.28		$$\underset{q\to 1}{lim}{\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\left(\genfrac{}{}{0.0pt}{}{n}{m}\right)=\frac{n!}{m!(n-m)!},$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E28 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

17.2.29		$${\left[\genfrac{}{}{0.0pt}{}{m+n}{m}\right]}_q=\frac{{(q^{n+1};q)}_m}{{(q;q)}_m},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E29 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

17.2.30		${\left[{\displaystyle \genfrac{}{}{0.0pt}{}{-n}{m}}\right]}_q$	$={\left[{\displaystyle \genfrac{}{}{0.0pt}{}{m+n-1}{m}}\right]}_q{(-1)}^m{q}^{-mn-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)},$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E30 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17
17.2.31		${\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n}{m}}\right]}_q$	$={\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n-1}{m-1}}\right]}_q+q^m{\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n-1}{m}}\right]}_q,$
ⓘ Symbols: ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E31 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17
17.2.32		${\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n}{m}}\right]}_q$	$={\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n-1}{m}}\right]}_q+q^{n-m}{\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n-1}{m-1}}\right]}_q,$
ⓘ Symbols: ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E32 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

17.2.33		$$\underset{n\to \mathrm{\infty }}{lim}{\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\frac{1}{{(q;q)}_m}=\frac{1}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E33 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

17.2.34		$$\underset{n\to \mathrm{\infty }}{lim}{\left[\genfrac{}{}{0.0pt}{}{rn+u}{sn+t}\right]}_q=\frac{1}{{(q;q)}_{\mathrm{\infty }}}=\prod _{j=1}^{\mathrm{\infty }}\frac{1}{(1-q^j)},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $r$: nonnegative integer and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E34 Encodings: TeX, pMML, png See also: Annotations for §17.2(ii), §17.2 and Ch.17

provided that $r>s$.

17.2.35		$$\sum _{j=0}^n{\left[\genfrac{}{}{0.0pt}{}{n}{j}\right]}_q{(-z)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}={(z;q)}_n=(1-z)(1-zq)\mathrm{\cdots }(1-z{q}^{n-1}).$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable Referenced by: §17.2(iii), §17.2(iii), §17.5 Permalink: http://dlmf.nist.gov/17.2.E35 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17

In the limit as $q\to 1$, (17.2.35) reduces to the standard binomial theorem

17.2.36		$$\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right){(-z)}^j={(1-z)}^n.$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/17.2.E36 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17

Also,

17.2.37		$$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q)}_n}{{(q;q)}_n}{z}^n=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}},$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $z$: complex variable Referenced by: §17.2(iii), §17.5 Permalink: http://dlmf.nist.gov/17.2.E37 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17

provided that . When $a=q^{m+1}$, where $m$ is a nonnegative integer, (17.2.37) reduces to the $q$-binomial series

17.2.38		${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n+m}{n}}\right]}_q{z}^n$	$={\displaystyle \frac{1}{{(z;q)}_{m+1}}}.$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable Referenced by: §17.2(iii) Permalink: http://dlmf.nist.gov/17.2.E38 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17
17.2.39		${\displaystyle \sum _{j=0}^n}{\left[{\displaystyle \genfrac{}{}{0.0pt}{}{n}{j}}\right]}_{q^2}{q}^j$	$={(-q;q)}_n,$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E39 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17
17.2.40		${\displaystyle \sum _{j=0}^{2n}}{(-1)}^j{\left[{\displaystyle \genfrac{}{}{0.0pt}{}{2n}{j}}\right]}_q$	$={(q;q^2)}_n.$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/17.2.E40 Encodings: TeX, pMML, png See also: Annotations for §17.2(iii), §17.2 and Ch.17

When $n\to \mathrm{\infty }$ in (17.2.35), and when $m\to \mathrm{\infty }$ in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)).

See also §26.9(ii).

The $q$-derivatives of $f(z)$ are defined by

17.2.41
ⓘ Defines: ${𝒟}_q$: $q$-differential operator (locally) Symbols: $q$: complex base and $z$: complex variable Referenced by: §17.2(iv), §18.27(i) Permalink: http://dlmf.nist.gov/17.2.E41 Encodings: TeX, pMML, png See also: Annotations for §17.2(iv), §17.2 and Ch.17

and

17.2.42
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator Referenced by: §17.2(iv) Permalink: http://dlmf.nist.gov/17.2.E42 Encodings: TeX, pMML, png See also: Annotations for §17.2(iv), §17.2 and Ch.17

When $q\to 1$ the $q$-derivatives converge to the corresponding ordinary derivatives.

If $f(x)$ is continuous at $x=0$, then

17.2.45		$${\int }_0^{1}f(x)d_{q}x=(1-q)\sum _{j=0}^{\mathrm{\infty }}f(q^j)q^j,$$
ⓘ Symbols: $\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.2.E45 Encodings: TeX, pMML, png See also: Annotations for §17.2(v), §17.2 and Ch.17

and more generally,

17.2.46		$${\int }_0^{a}f(x)d_{q}x=a(1-q)\sum _{j=0}^{\mathrm{\infty }}f(a{q}^j)q^j.$$
ⓘ Symbols: $\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.2.E46 Encodings: TeX, pMML, png See also: Annotations for §17.2(v), §17.2 and Ch.17

If $f(x)$ is continuous on $[0,a]$, then

17.2.47		$$\underset{q\to 1-}{lim}{\int }_0^{a}f(x)d_{q}x={\int }_0^{a}f(x)dx.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable Permalink: http://dlmf.nist.gov/17.2.E47 Encodings: TeX, pMML, png See also: Annotations for §17.2(v), §17.2 and Ch.17

17.2.49		$$1+\sum _{n=1}^{\mathrm{\infty }}\frac{q^{n^2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^n)}=\prod _{n=0}^{\mathrm{\infty }}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})},$$
ⓘ Symbols: $q$: complex base and $n$: nonnegative integer Referenced by: §17.14 Permalink: http://dlmf.nist.gov/17.2.E49 Encodings: TeX, pMML, png See also: Annotations for §17.2(vi), §17.2 and Ch.17

17.2.50		$$1+\sum _{n=1}^{\mathrm{\infty }}\frac{q^{n^2+n}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^n)}=\prod _{n=0}^{\mathrm{\infty }}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}.$$
ⓘ Symbols: $q$: complex base and $n$: nonnegative integer Referenced by: §17.14 Permalink: http://dlmf.nist.gov/17.2.E50 Encodings: TeX, pMML, png See also: Annotations for §17.2(vi), §17.2 and Ch.17

These identities are the first in a large collection of similar results. See §17.14.