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17 q-Hypergeometric and Related FunctionsProperties

Β§17.2 Calculus

Contents
  1. Β§17.2(i) q-Calculus
  2. Β§17.2(ii) q-Binomial Coefficients
  3. Β§17.2(iii) q-Binomial Theorem
  4. Β§17.2(iv) q-Derivatives
  5. Β§17.2(v) q-Integrals
  6. Β§17.2(vi) Rogers–Ramanujan Identities

Β§17.2(i) q-Calculus

β“˜
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:
Β§26.9(ii), Β§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 Β§27.14(ii) were added below (17.2.6).
See also:
Annotations for Β§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}),

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),

when this product converges.

17.2.4 \left(a;q\right)_{\infty}=\prod_{j=0}^{\infty}(1-aq^{j}),

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 Β§27.14. Let q={\mathrm{e}}^{-t} and \hat{q}={\mathrm{e}}^{\ifrac{-4\pi^{2}}{t}}. Then

17.2.6_1 \left(q;q\right)_{\infty}=\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,
17.2.6_2 \left(-q;q\right)_{\infty}=\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.
β“˜
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:
Β§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 Β§17.2(i), Β§17.2 and Ch.17

For these and similar results see Apostol (1990, Ch.Β 3) and Katsurada (2003, Β§3). Note that (17.2.6_1) is just (27.14.14) with a=d=0 and -b=c=1.

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},
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},
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}.

more generally,

17.2.21 \left(a^{2};q^{2}\right)_{n}=\left(a;q\right)_{n}\left(-a;q\right)_{n},
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},
β“˜
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 Β§17.2(i), Β§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},
β“˜
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 Β§17.2(i), Β§17.2 and Ch.17

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

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},

Β§17.2(ii) q-Binomial Coefficients

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

provided that r>s.

Β§17.2(iii) q-Binomial Theorem

β“˜
Keywords:
q-binomial theorem
Notes:
See Andrews (1976, pp.Β 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 β€œBinomial Theorem”) to clarify that it concerns specifically q-generalizations.
Addition (effective with 1.0.7):
The cross-reference to Β§26.9(ii) has been added at the end of this subsection.

Suggested 2013-11-25 by Howard Cohl

See also:
Annotations for Β§17.2 and Ch.17

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

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)).

See also Β§26.9(ii).

Β§17.2(iv) q-Derivatives

β“˜
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 β€œDerivatives”) to clarify that it concerns specifically q-generalizations.
See also:
Annotations for Β§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}
β“˜
Defines:
\mathcal{D}_{q}: q-differential operator (locally)
Symbols:
q: complex base and z: complex variable
Referenced by:
Β§17.2(iv), Β§18.27(i)
Permalink:
http://dlmf.nist.gov/17.2.E41
Encodings:
TeX, pMML, png
See also:
Annotations for Β§17.2(iv), Β§17.2 and Ch.17

and

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

Product Rule

β“˜
See also:
Annotations for Β§17.2(iv), Β§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).
β“˜
Symbols:
q: complex base, z: complex variable and \mathcal{D}_{q}: q-differential operator
Referenced by:
Β§17.2(iv)
Permalink:
http://dlmf.nist.gov/17.2.E43
Encodings:
TeX, pMML, png
See also:
Annotations for Β§17.2(iv), Β§17.2(iv), Β§17.2 and Ch.17

Leibniz Rule

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

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

Β§17.2(v) q-Integrals

β“˜
Defines:
\,{\mathrm{d}}_{\NVar{q}}\NVar{x}: q-differential of x and \qint: q-integral
Keywords:
q-integrals
Referenced by:
Β§18.27(i), Β§5.18(iii)
Permalink:
http://dlmf.nist.gov/17.2.v
Errata (effective with 1.2.5):
This section was renamed (from β€œIntegrals”) 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 Β§17.2 and Ch.17

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

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}.

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

Infinite Range

β“˜
See also:
Annotations for Β§17.2(v), Β§17.2 and Ch.17

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

Β§17.2(vi) Rogers–Ramanujan Identities

β“˜
Keywords:
Rogers–Ramanujan identities, q-calculus
Notes:
See Andrews (1976, Β§7.3).
Referenced by:
Β§17.12, Β§26.10(iv)
Permalink:
http://dlmf.nist.gov/17.2.vi
See also:
Annotations for Β§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})},
β“˜
Symbols:
q: complex base and n: nonnegative integer
Referenced by:
Β§17.14
Permalink:
http://dlmf.nist.gov/17.2.E49
Encodings:
TeX, pMML, png
See also:
Annotations for Β§17.2(vi), Β§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})}.
β“˜
Symbols:
q: complex base and n: nonnegative integer
Referenced by:
Β§17.14
Permalink:
http://dlmf.nist.gov/17.2.E50
Encodings:
TeX, pMML, png
See also:
Annotations for Β§17.2(vi), Β§17.2 and Ch.17

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

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