-Hypergeometric and Related Functions Properties 17.1 Special Notation 17.3

-Special Functions

§17.2 Calculus

Keywords:: -calculus
Referenced by:: §18.27(i), §18.28(i), ¶ ‣ §18.33(iv), §27.14(vi)
Permalink:: http://dlmf.nist.gov/17.2

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

§17.2(i) -Calculus

Defines:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial)
Referenced by:: §26.9(ii), §5.18(i)
Permalink:: http://dlmf.nist.gov/17.2.i

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

17.2.1 $\left(a;q\right)_{{n}}=(1-a)(1-aq)\cdots(1-aq^{{n-1}}),$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E1
Encodings:: TeX, pMML, png

17.2.2 $\left(a;q\right)_{{-n}}=\frac{1}{\left(aq^{{-n}};q\right)_{{n}}}=\frac{(-q/a)^% {n}q^{{\binom{n}{2}}}}{\left(q/a;q\right)_{{n}}}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E2
Encodings:: TeX, pMML, png

For $\nu\in\Complex$

17.2.3 $\left(a;q\right)_{{\nu}}=\prod_{{j=0}}^{\infty}\left(\frac{1-aq^{j}}{1-aq^{{% \nu+j}}}\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E3
Encodings:: TeX, pMML, png

when this product converges.

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

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E4
Encodings:: TeX, pMML, png

17.2.5 $\left(a_{1},a_{2},\dots,a_{r};q\right)_{{n}}=\prod_{{j=1}}^{r}\left(a_{j};q% \right)_{{n}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E5
Encodings:: TeX, pMML, png

17.2.6 $\left(a_{1},a_{2},\dots,a_{r};q\right)_{{\infty}}=\prod_{{j=1}}^{r}\left(a_{j}% ;q\right)_{{\infty}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E6
Encodings:: TeX, pMML, png

17.2.7 $\left(a;q^{{-1}}\right)_{{n}}=\left(a^{{-1}};q\right)_{{n}}(-a)^{n}q^{{-\binom% {n}{2}}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E7
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E8
Encodings:: TeX, pMML, png

17.2.9 $\left(a;q\right)_{{n}}=\left(q^{{1-n}}/a;q\right)_{{n}}(-a)^{n}q^{{\binom{n}{2% }}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E9
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E10
Encodings:: TeX, pMML, png

17.2.11 $\left(aq^{{-n}};q\right)_{{n}}=\left(q/a;q\right)_{{n}}\left(-\frac{a}{q}% \right)^{n}q^{{-\binom{n}{2}}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E11
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E12
Encodings:: TeX, pMML, png

17.2.13 $\left(a;q\right)_{{n-k}}=\frac{\left(a;q\right)_{{n}}}{\left(q^{{1-n}}/a;q% \right)_{{k}}}\left(-\frac{q}{a}\right)^{k}q^{{\binom{k}{2}-nk}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E13
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E14
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E15
Encodings:: TeX, pMML, png

17.2.16 $\left(aq^{{-n}};q\right)_{{n-k}}=\frac{\left(q/a;q\right)_{{n}}}{\left(q/a;q% \right)_{{k}}}\left(-\frac{a}{q}\right)^{{n-k}}q^{{\binom{k}{2}-\binom{n}{2}}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E16
Encodings:: TeX, pMML, png

17.2.17 $\left(aq^{n};q\right)_{{k}}=\frac{\left(a;q\right)_{{k}}\left(aq^{k};q\right)_% {{n}}}{\left(a;q\right)_{{n}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E17
Encodings:: TeX, pMML, png

17.2.18 $\left(aq^{k};q\right)_{{n-k}}=\frac{\left(a;q\right)_{{n}}}{\left(a;q\right)_{% {k}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E18
Encodings:: TeX, pMML, png

17.2.19 $\left(a;q\right)_{{2n}}=\left(a,aq;q^{2}\right)_{{n}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E19
Encodings:: TeX, pMML, png

more generally,

17.2.20 $\left(a;q\right)_{{kn}}=\left(a,aq,\dots,aq^{{k-1}};q^{k}\right)_{{n}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E20
Encodings:: TeX, pMML, png

17.2.21 $\left(a^{2};q^{2}\right)_{{n}}=\left(a;q\right)_{{n}}\left(-a;q\right)_{{n}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E21
Encodings:: TeX, pMML, png

17.2.22 $\frac{\left(qa^{{\frac{1}{2}}},-aq^{{\frac{1}{2}}};q\right)_{{n}}}{\left(a^{{% \frac{1}{2}}},-a^{{\frac{1}{2}}};q\right)_{{n}}}=\frac{\left(aq^{2};q^{2}% \right)_{{n}}}{\left(a;q^{2}\right)_{{n}}}=\frac{1-aq^{{2n}}}{1-a},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E22
Encodings:: TeX, pMML, png

more generally,

17.2.23 $\frac{\left(aq^{{\frac{1}{k}}},q\omega_{k}a^{{\frac{1}{k}}},\dots,q\omega_{k}^% {{k-1}}a^{{\frac{1}{k}}};q\right)_{{n}}}{\left(a^{{\frac{1}{k}}},\omega_{k}a^{% {\frac{1}{k}}},\dots,\omega_{k}^{{k-1}}a^{{\frac{1}{k}}};q\right)_{{n}}}=\frac% {\left(aq^{k};q^{k}\right)_{{n}}}{\left(a;q^{k}\right)_{{n}}}=\frac{1-aq^{{kn}% }}{1-a},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E23
Encodings:: TeX, pMML, png

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

17.2.24 $\lim_{{\tau\to 0}}\left(a/\tau;q\right)_{{n}}\tau^{n}=\lim_{{\sigma\to\infty}}% \left(a\sigma;q\right)_{{n}}\sigma^{{-n}}=(-a)^{n}q^{{\binom{n}{2}}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E24
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E25
Encodings:: TeX, pMML, png

17.2.26 $\lim_{{\tau\to 0}}\frac{\left(a/\tau;q\right)_{{n}}\left(b/\tau;q\right)_{{n}}% }{\left(c/\tau^{2};q\right)_{{n}}}=(-1)^{n}\left(\frac{ab}{c}\right)^{n}q^{{% \binom{n}{2}}}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E26
Encodings:: TeX, pMML, png

§17.2(ii) Binomial Coefficients

Keywords:: -binomial coefficient
Referenced by:: §26.9(ii)
Permalink:: http://dlmf.nist.gov/17.2.ii

17.2.27 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}=\frac{\left(q;q\right)_{{n}}}{\left(q;q% \right)_{{m}}\left(q;q\right)_{{n-m}}}\\ =\frac{\left(q^{{-n}};q\right)_{{m}}(-1)^{m}q^{{nm-\binom{m}{2}}}}{\left(q;q% \right)_{{m}}},$

Defines:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial)
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E27
Encodings:: TeX, pMML, png

17.2.28 $\lim_{{q\to 1}}\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}=\binom{n}{m}=\frac{n!}{m!(n% -m)!},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : : factorial, $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E28
Encodings:: TeX, pMML, png

17.2.29 $\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{{q}}=\frac{\left(q^{{n+1}};q\right)_{{m}}}{% \left(q;q\right)_{{m}}},$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E29
Encodings:: TeX, pMML, png

17.2.30 $\genfrac{[}{]}{0.0pt}{}{-n}{m}_{{q}}=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{{q}}(-% 1)^{m}q^{{-mn-\binom{m}{2}}},$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E30
Encodings:: TeX, pMML, png

17.2.31 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}=\genfrac{[}{]}{0.0pt}{}{n-1}{m-1}_{{q}}+q^% {m}\genfrac{[}{]}{0.0pt}{}{n-1}{m}_{{q}},$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E31
Encodings:: TeX, pMML, png

17.2.32 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}=\genfrac{[}{]}{0.0pt}{}{n-1}{m}_{{q}}+q^{{% n-m}}\genfrac{[}{]}{0.0pt}{}{n-1}{m-1}_{{q}},$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E32
Encodings:: TeX, pMML, png

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:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E33
Encodings:: TeX, pMML, png

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:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E34
Encodings:: TeX, pMML, png

provided that .

§17.2(iii) Binomial Theorem

Notes:: See Andrews (1976, pp. 17, 36, 37, and 49).
Keywords:: -binomial theorem
Permalink:: http://dlmf.nist.gov/17.2.iii

17.2.35 $\sum_{{j=0}}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{{q}}(-z)^{j}q^{{\binom{j}{2}}}=% \left(z;q\right)_{{n}}={(1-z)(1-zq)\cdots(1-zq^{{n-1}})}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer and : complex variable
Referenced by:: §17.2(iii), §17.2(iii), ¶ ‣ §17.5
Permalink:: http://dlmf.nist.gov/17.2.E35
Encodings:: TeX, pMML, png

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

17.2.36 $\sum_{{j=0}}^{n}\binom{n}{j}(-z)^{j}=(1-z)^{n}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : nonnegative integer, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/17.2.E36
Encodings:: TeX, pMML, png

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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : complex variable
Referenced by:: §17.2(iii), ¶ ‣ §17.5
Permalink:: http://dlmf.nist.gov/17.2.E37
Encodings:: TeX, pMML, png

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

17.2.38 $\sum_{{n=0}}^{\infty}\genfrac{[}{]}{0.0pt}{}{n+m}{n}_{{q}}z^{n}=\frac{1}{\left% (z;q\right)_{{m+1}}}.$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer and : complex variable
Referenced by:: §17.2(iii)
Permalink:: http://dlmf.nist.gov/17.2.E38
Encodings:: TeX, pMML, png

17.2.39 $\sum_{{j=0}}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{{q^{2}}}q^{j}=\left(-q;q\right)% _{{n}},$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E39
Encodings:: TeX, pMML, png

17.2.40 $\sum_{{j=0}}^{{2n}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{2n}{j}_{{q}}=\left(q;q^{2}% \right)_{{n}}.$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E40
Encodings:: TeX, pMML, png

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) Derivatives

Notes:: (17.2.43) is derived from (17.2.41); (17.2.42) and (17.2.44) follow by induction.
Keywords:: -derivatives
Permalink:: http://dlmf.nist.gov/17.2.iv

The -derivatives of are defined by

17.2.41 $\mathcal{D}_{q}f(z)=\begin{cases}\dfrac{f(z)-f(zq)}{(1-q)z},&z\neq 0,\\ f^{{\prime}}(0),&z=0,\end{cases}$

Symbols:: : complex base and : complex variable
Referenced by:: §17.2(iv), §18.27(i)
Permalink:: http://dlmf.nist.gov/17.2.E41
Encodings:: TeX, pMML, png

and

17.2.42 $f^{{[n]}}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{{-n}}(1-q)^{{-n}}\sum_{{j% =0}}^{n}q^{{-nj+\binom{j+1}{2}}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{{q}}f(% zq^{j}),&z\neq 0,\\ \dfrac{f^{{(n)}}(0)\left(q;q\right)_{{n}}}{n!(1-q)^{n}},&z=0.\end{cases}$