17 q-Hypergeometric and Related Functions Properties 17.1 Special Notation 17.3

q

q

-Special Functions

§17.2 Calculus

ⓘ

Keywords:: $q$ -calculus
Referenced by:: §18.27(i), §18.28(i), §18.33(iv), §27.14(vi), Erratum (V1.2.5) for Chapter 17 Additions
Permalink:: http://dlmf.nist.gov/17.2
See also:: Annotations for Ch.17

§17.2(i) $q$ -Calculus
§17.2(ii) $q$ -Binomial Coefficients
§17.2(iii) $q$ -Binomial Theorem
§17.2(iv) $q$ -Derivatives
§17.2(v) $q$ -Integrals
§17.2(vi) Rogers–Ramanujan Identities

§17.2(i) $q$ -Calculus

ⓘ

Defines:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol
Referenced by:: §26.9(ii), §5.18(i), Erratum (V1.0.28) for Subsection 17.2(i)
Permalink:: http://dlmf.nist.gov/17.2.i
Addition (effective with 1.1.5):: Equations (17.2.6_1), (17.2.6_2) were added.
Addition (effective with 1.0.28):: A sentence and reference to §27.14(ii) were added below (17.2.6).
See also:: Annotations for §17.2 and Ch.17

For $n=0,1,2,\dots$ ,

17.2.1		$\left(a;q\right)_{n}=(1-a)(1-aq)\cdots(1-aq^{n-1}),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(a;q\right)_{-n}=\frac{1}{\left(aq^{-n};q\right)_{n}}=\frac{(-q/a)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}}{\left(q/a;q\right)_{n}}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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 $\nu\in\mathbb{C}$

17.2.3		$\left(a;q\right)_{\nu}=\prod_{j=0}^{\infty}\left(\frac{1-aq^{j}}{1-aq^{\nu+j}}% \right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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	$\displaystyle\left(a;q\right)_{\infty}$	$\displaystyle=\prod_{j=0}^{\infty}(1-aq^{j}),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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	$\displaystyle\left(a_{1},a_{2},\dots,a_{r};q\right)_{n}$	$\displaystyle=\prod_{j=1}^{r}\left(a_{j};q\right)_{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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	$\displaystyle\left(a_{1},a_{2},\dots,a_{r};q\right)_{\infty}$	$\displaystyle=\prod_{j=1}^{r}\left(a_{j};q\right)_{\infty}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\mathit{f}\left(q\right)=q^{\ifrac{-1}{24}}\eta\left(\frac{\ln q}{2\pi\mathrm{% i}}\right)=\left(q;q\right)_{\infty}$ see §27.14. Let $q={\mathrm{e}}^{-t}$ and $\hat{q}={\mathrm{e}}^{\ifrac{-4\pi^{2}}{t}}$ . Then
17.2.6_1	$\displaystyle\left(q;q\right)_{\infty}$	$\displaystyle=\sqrt{\frac{2\pi}{t}}\exp\left(-\frac{{\pi}^{2}}{6t}+\frac{t}{24% }\right)\left(\hat{q};\hat{q}\right)_{\infty},$
$\Re t>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\Re$ : real part and $q$ : complex base Referenced by: §17.2(i), §17.2(i), Erratum (V1.1.5) for Additions Permalink: http://dlmf.nist.gov/17.2.E6_1 Encodings: TeX, pMML, png Addition (effective with 1.1.5): This equation was added. See also: Annotations for §17.2(i), §17.2 and Ch.17
17.2.6_2	$\displaystyle\left(-q;q\right)_{\infty}$	$\displaystyle=\frac{1}{\sqrt{2}}\exp\left(\frac{{\pi}^{2}}{12t}+\frac{t}{24}% \right)\left(\hat{q}^{\frac{1}{2}};\hat{q}\right)_{\infty},$
$t>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $q$ : complex base Referenced by: §17.2(i), Erratum (V1.1.5) for Additions Permalink: http://dlmf.nist.gov/17.2.E6_2 Encodings: TeX, pMML, png Addition (effective with 1.1.5): This equation was added. See also: Annotations for §17.2(i), §17.2 and Ch.17
For these and similar results see Apostol (1990, Ch. 3) and Katsurada (2003, §3). Note that (17.2.6_1) is just (27.14.14) with $a=d=0$ and $-b=c=1$ .

17.2.7		$\left(a;q^{-1}\right)_{n}=\left(a^{-1};q\right)_{n}(-a)^{n}q^{-\genfrac{(}{)}{% 0.0pt}{}{n}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\frac{\left(a;q^{-1}\right)_{n}}{\left(b;q^{-1}\right)_{n}}=\frac{\left(a^{-1}% ;q\right)_{n}}{\left(b^{-1};q\right)_{n}}\left(\frac{a}{b}\right)^{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(a;q\right)_{n}=\left(q^{1-n}/a;q\right)_{n}(-a)^{n}q^{\genfrac{(}{)}{0.0% pt}{}{n}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\frac{\left(a;q\right)_{n}}{\left(b;q\right)_{n}}=\frac{\left(q^{1-n}/a;q% \right)_{n}}{\left(q^{1-n}/b;q\right)_{n}}\left(\frac{a}{b}\right)^{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(aq^{-n};q\right)_{n}=\left(q/a;q\right)_{n}\left(-\frac{a}{q}\right)^{n}% q^{-\genfrac{(}{)}{0.0pt}{}{n}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\frac{\left(aq^{-n};q\right)_{n}}{\left(bq^{-n};q\right)_{n}}=\frac{\left(q/a;% q\right)_{n}}{\left(q/b;q\right)_{n}}\left(\frac{a}{b}\right)^{n}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(a;q\right)_{n-k}=\frac{\left(a;q\right)_{n}}{\left(q^{1-n}/a;q\right)_{k% }}\left(-\frac{q}{a}\right)^{k}q^{\genfrac{(}{)}{0.0pt}{}{k}{2}-nk},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\frac{\left(a;q\right)_{n-k}}{\left(b;q\right)_{n-k}}=\frac{\left(a;q\right)_{% n}}{\left(b;q\right)_{n}}\frac{\left(q^{1-n}/b;q\right)_{k}}{\left(q^{1-n}/a;q% \right)_{k}}\left(\frac{b}{a}\right)^{k},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(aq^{-n};q\right)_{k}=\frac{\left(a;q\right)_{k}\left(q/a;q\right)_{n}}{% \left(q^{1-k}/a;q\right)_{n}}q^{-nk},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(aq^{-n};q\right)_{n-k}=\frac{\left(q/a;q\right)_{n}}{\left(q/a;q\right)_% {k}}\left(-\frac{a}{q}\right)^{n-k}q^{\genfrac{(}{)}{0.0pt}{}{k}{2}-\genfrac{(% }{)}{0.0pt}{}{n}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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	$\displaystyle\left(aq^{n};q\right)_{k}$	$\displaystyle=\frac{\left(a;q\right)_{k}\left(aq^{k};q\right)_{n}}{\left(a;q% \right)_{n}},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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	$\displaystyle\left(aq^{k};q\right)_{n-k}$	$\displaystyle=\frac{\left(a;q\right)_{n}}{\left(a;q\right)_{k}}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\left(a;q\right)_{2n}=\left(a,aq;q^{2}\right)_{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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		$\left(a;q\right)_{kn}=\left(a,aq,\dots,aq^{k-1};q^{k}\right)_{n}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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		$\left(a^{2};q^{2}\right)_{n}=\left(a;q\right)_{n}\left(-a;q\right)_{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\frac{\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q\right)_{n}}{\left(a^{\frac{1}% {2}},-a^{\frac{1}{2}};q\right)_{n}}=\frac{\left(aq^{2};q^{2}\right)_{n}}{\left% (a;q^{2}\right)_{n}}=\frac{1-aq^{2n}}{1-a},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q\right)_{n}$ instead of $\left(qa^{\frac{1}{2}},-aq^{\frac{1}{2}};q\right)_{n}$ . Reported 2017-06-26 by Jason Zhao See also: Annotations for §17.2(i), §17.2 and Ch.17

more generally,

17.2.23		$\frac{\left(qa^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1% }a^{\frac{1}{k}};q\right)_{n}}{\left(a^{\frac{1}{k}},\omega_{k}a^{\frac{1}{k}}% ,\dots,\omega_{k}^{k-1}a^{\frac{1}{k}};q\right)_{n}}=\frac{\left(aq^{k};q^{k}% \right)_{n}}{\left(a;q^{k}\right)_{n}}=\frac{1-aq^{kn}}{1-a},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left(qa^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1}a^{% \frac{1}{k}};q\right)_{n}$ instead of $\left(aq^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1}a^{% \frac{1}{k}};q\right)_{n}$ . Reported 2017-06-26 by Jason Zhao See also: Annotations for §17.2(i), §17.2 and Ch.17

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

17.2.24		$\lim_{\tau\to 0}\left(a/\tau;q\right)_{n}\tau^{n}=\lim_{\sigma\to\infty}\left(% a\sigma;q\right)_{n}\sigma^{-n}=(-a)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\lim_{\tau\to 0}\frac{\left(a/\tau;q\right)_{n}}{\left(b/\tau;q\right)_{n}}=% \lim_{\sigma\to\infty}\frac{\left(a\sigma;q\right)_{n}}{\left(b\sigma;q\right)% _{n}}=\left(\frac{a}{b}\right)^{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\lim_{\tau\to 0}\frac{\left(a/\tau;q\right)_{n}\left(b/\tau;q\right)_{n}}{% \left(c/\tau^{2};q\right)_{n}}=(-1)^{n}\left(\frac{ab}{c}\right)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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(ii) $q$ -Binomial Coefficients

ⓘ

Keywords:: $q$ -binomial coefficient
Referenced by:: §26.9(ii)
Permalink:: http://dlmf.nist.gov/17.2.ii
Errata (effective with 1.2.5):: This section was renamed (from “Binomial Coefficients”) to clarify that it concerns specifically $q$ -generalizations.
See also:: Annotations for §17.2 and Ch.17

17.2.27		$\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)% _{m}\left(q;q\right)_{n-m}}\\ =\frac{\left(q^{-n};q\right)_{m}(-1)^{m}q^{nm-\genfrac{(}{)}{0.0pt}{}{m}{2}}}{% \left(q;q\right)_{m}},$
ⓘ Defines: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial) Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{(}{)}{0.0pt}{}{n}{m}=% \frac{n!}{m!(n-m)!},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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		$\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{q}=\frac{\left(q^{n+1};q\right)_{m}}{\left(q;% q\right)_{m}},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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	$\displaystyle\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}$	$\displaystyle=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{q}(-1)^{m}q^{-mn-\genfrac{(}{% )}{0.0pt}{}{m}{2}},$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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	$\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}$	$\displaystyle=\genfrac{[}{]}{0.0pt}{}{n-1}{m-1}_{q}+q^{m}\genfrac{[}{]}{0.0pt}% {}{n-1}{m}_{q},$
ⓘ Symbols: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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	$\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}$	$\displaystyle=\genfrac{[}{]}{0.0pt}{}{n-1}{m}_{q}+q^{n-m}\genfrac{[}{]}{0.0pt}% {}{n-1}{m-1}_{q},$
ⓘ Symbols: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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		$\lim_{n\to\infty}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{1}{\left(q;q\right)_{% m}}=\frac{1}{(1-q)(1-q^{2})\cdots(1-q^{m})},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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		$\lim_{n\to\infty}\genfrac{[}{]}{0.0pt}{}{rn+u}{sn+t}_{q}=\frac{1}{\left(q;q% \right)_{\infty}}=\prod_{j=1}^{\infty}\frac{1}{(1-q^{j})},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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(iii) $q$ -Binomial Theorem

ⓘ

Keywords:: $q$ -binomial theorem
Notes:: See Andrews (1976, pp. 17, 36, 37, and 49).
Referenced by:: Erratum (V1.0.7) for Usability
Permalink:: http://dlmf.nist.gov/17.2.iii
Errata (effective with 1.2.5):: This section was renamed (from “Binomial Theorem”) to clarify that it concerns specifically $q$ -generalizations.
Addition (effective with 1.0.7):: The cross-reference to §26.9(ii) has been added at the end of this subsection.
Suggested 2013-11-25 by Howard Cohl
See also:: Annotations for §17.2 and Ch.17

17.2.35		$\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}(-z)^{j}q^{\genfrac{(}{)}{0.0pt% }{}{j}{2}}=\left(z;q\right)_{n}={(1-z)(1-zq)\cdots(1-zq^{n-1})}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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		$\sum_{j=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}(-z)^{j}=(1-z)^{n}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}}{\left(q;q\right)_{n}}z^{n}=% \frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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 $\left|z\right|<1$ . When $a=q^{m+1}$ , where $m$ is a nonnegative integer, (17.2.37) reduces to the $q$ -binomial series

17.2.38	$\displaystyle\sum_{n=0}^{\infty}\genfrac{[}{]}{0.0pt}{}{n+m}{n}_{q}z^{n}$	$\displaystyle=\frac{1}{\left(z;q\right)_{m+1}}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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	$\displaystyle\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q^{2}}q^{j}$	$\displaystyle=\left(-q;q\right)_{n},$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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	$\displaystyle\sum_{j=0}^{2n}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{2n}{j}_{q}$	$\displaystyle=\left(q;q^{2}\right)_{n}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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 $n\to\infty$ in (17.2.35), and when $m\to\infty$ in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)).

§17.2(iv) $q$ -Derivatives

ⓘ

Keywords:: $q$ -derivatives
Notes:: (17.2.43) is derived from (17.2.41); (17.2.42) and (17.2.44) follow by induction.
Permalink:: http://dlmf.nist.gov/17.2.iv
Errata (effective with 1.2.5):: This section was renamed (from “Derivatives”) to clarify that it concerns specifically $q$ -generalizations.
See also:: Annotations for §17.2 and Ch.17

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

17.2.41		$\mathcal{D}_{q}f(z)=\begin{cases}\dfrac{f(z)-f(zq)}{(1-q)z},&z\neq 0,\\ f^{\prime}(0),&z=0,\end{cases}$
ⓘ Defines: $\mathcal{D}_{q}$ : $q$ -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		$f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}% q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{% q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -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.

Product Rule

ⓘ

See also:: Annotations for §17.2(iv), §17.2 and Ch.17

17.2.43		$\mathcal{D}_{q}(f(z)g(z))=g(z)f^{[1]}(z)+f(zq)g^{[1]}(z).$
ⓘ Symbols: $q$ : complex base, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator Referenced by: §17.2(iv) Permalink: http://dlmf.nist.gov/17.2.E43 Encodings: TeX, pMML, png See also: Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

Leibniz Rule

ⓘ

Keywords:: $q$ -Leibniz rule
See also:: Annotations for §17.2(iv), §17.2 and Ch.17

17.2.44		$\mathcal{D}_{q}^{n}(f(z)g(z))=\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}f% ^{[n-j]}(zq^{j})g^{[j]}(z).$
ⓘ Symbols: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator Referenced by: §17.2(iv) Permalink: http://dlmf.nist.gov/17.2.E44 Encodings: TeX, pMML, png See also: Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

$q$ -differential equations are considered in §17.6(iv).

§17.2(v) $q$ -Integrals

ⓘ

Defines:: $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of $x$ and $\qint$ : q-integral
Keywords:: $q$ -integrals
Referenced by:: §18.27(i), §5.18(iii)
Permalink:: http://dlmf.nist.gov/17.2.v
Errata (effective with 1.2.5):: This section was renamed (from “Integrals”) to clarify that it concerns specifically $q$ -generalizations. The generalization of $q$ -integral limits to $[a,b]$ was added after (17.2.45).
See also:: Annotations for §17.2 and Ch.17

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

17.2.45		$\qint_{0}^{1}f(x)\,{\mathrm{d}}_{q}x=(1-q)\sum_{j=0}^{\infty}f(q^{j})q^{j},$
ⓘ Symbols: $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of $x$ , $\qint$ : q-integral, $q$ : complex base, $j$ : nonnegative integer and $x$ : real variable Referenced by: §17.2(v), Erratum (V1.2.5) for Chapter 17 Additions 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, $\qint_{a}^{b}f(x)\,{\mathrm{d}}_{q}x=\qint_{0}^{b}f(x)\,{\mathrm{d}}_{q}x-% \qint_{0}^{a}f(x)\,{\mathrm{d}}_{q}x$ , where

17.2.46		$\qint_{0}^{a}f(x)\,{\mathrm{d}}_{q}x=a(1-q)\sum_{j=0}^{\infty}f(aq^{j})q^{j}.$
ⓘ Symbols: $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of $x$ , $\qint$ : q-integral, $q$ : 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		$\lim_{q\to 1-}\qint_{0}^{a}f(x)\,{\mathrm{d}}_{q}x=\int_{0}^{a}f(x)\,\mathrm{d% }x.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of $x$ , $\qint$ : q-integral, $q$ : 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

Infinite Range

ⓘ

See also:: Annotations for §17.2(v), §17.2 and Ch.17

17.2.48		$\qint_{0}^{\infty}f(x)\,{\mathrm{d}}_{q}x=\lim_{n\to\infty}\qint_{0}^{q^{-n}}f% (x)\,{\mathrm{d}}_{q}x=(1-q)\sum_{j=-\infty}^{\infty}f(q^{j})q^{j},$
ⓘ Symbols: $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of $x$ , $\qint$ : q-integral, $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/17.2.E48 Encodings: TeX, pMML, png See also: Annotations for §17.2(v), §17.2(v), §17.2 and Ch.17

provided that $\sum_{j=-\infty}^{\infty}f(q^{j})q^{j}$ converges.

§17.2(vi) Rogers–Ramanujan Identities

ⓘ

Keywords:: Rogers–Ramanujan identities, $q$ -calculus
Notes:: See Andrews (1976, §7.3).
Referenced by:: §17.12, §26.10(iv)
Permalink:: http://dlmf.nist.gov/17.2.vi
See also:: Annotations for §17.2 and Ch.17

17.2.49		$1+\sum_{n=1}^{\infty}\frac{q^{n^{2}}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=% 0}^{\infty}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})},$
ⓘ 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		$1+\sum_{n=1}^{\infty}\frac{q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{% n=0}^{\infty}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}.$
ⓘ 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