# Β§17.3 $q$-Elementary and $q$-Special Functions

## Β§17.3(i) Elementary Functions

### $q$-Exponential Functions

 17.3.1 $e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$ β Defines: $e_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable Referenced by: Β§17.5 Permalink: http://dlmf.nist.gov/17.3.E1 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17
 17.3.2 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$ β Defines: $E_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function 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, $n$: nonnegative integer and $x$: real variable Referenced by: Β§17.5 Permalink: http://dlmf.nist.gov/17.3.E2 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17

### $q$-Sine Functions

 17.3.3 $\operatorname{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left% (q;q\right)_{2n+1}},$ β Defines: $\operatorname{sin}_{\NVar{q}}\left(\NVar{x}\right)$: $q$-sine function Symbols: $\mathrm{i}$: imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.3.E3 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17
 17.3.4 $\operatorname{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2% n+1}}{\left(q;q\right)_{2n+1}}.$ β Defines: $\operatorname{Sin}_{\NVar{q}}\left(\NVar{x}\right)$: $q$-sine function Symbols: $\mathrm{i}$: imaginary unit, $E_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.3.E4 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17

### $q$-Cosine Functions

 17.3.5 $\operatorname{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}\left(ix\right)+e_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q% \right)_{2n}},$ β Defines: $\operatorname{cos}_{\NVar{q}}\left(\NVar{x}\right)$: $q$-cosine function Symbols: $\mathrm{i}$: imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.3.E5 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17
 17.3.6 $\operatorname{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}% }{\left(q;q\right)_{2n}}.$ β Defines: $\operatorname{Cos}_{\NVar{q}}\left(\NVar{x}\right)$: $q$-cosine function Symbols: $\mathrm{i}$: imaginary unit, $E_{\NVar{q}}\left(\NVar{x}\right)$: $q$-exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.3.E6 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(i), Β§17.3(i), Β§17.3 and Ch.17

See Β§5.18.

## Β§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

### $q$-Bernoulli Polynomials

 17.3.7 $\beta_{n}\left(x,q\right)=(1-q)^{1-n}\sum_{r=0}^{n}(-1)^{r}\genfrac{(}{)}{0.0% pt}{}{n}{r}\frac{r+1}{(1-q^{r+1})}q^{rx}.$ β Defines: $\beta_{\NVar{n}}\left(\NVar{x},\NVar{q}\right)$: $q$-Bernoulli polynomial Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/17.3.E7 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(iii), Β§17.3(iii), Β§17.3 and Ch.17

### $q$-Euler Numbers

 17.3.8 $A_{m,s}\left(q\right)=q^{\genfrac{(}{)}{0.0pt}{}{s-m}{2}+\genfrac{(}{)}{0.0pt}% {}{s}{2}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}% {0.0pt}{}{m+1}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$ β Defines: $A_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$: $q$-Euler number 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, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.3.E8 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(iii), Β§17.3(iii), Β§17.3 and Ch.17

### $q$-Stirling Numbers

 17.3.9 $a_{m,s}\left(q\right)=\frac{q^{-\genfrac{(}{)}{0.0pt}{}{s}{2}}(1-q)^{s}}{\left% (q;q\right)_{s}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}% \genfrac{[}{]}{0.0pt}{}{s}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$ β Defines: $a_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$: $q$-Stirling number 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, $m$: nonnegative integer and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.3.E9 Encodings: TeX, pMML, png See also: Annotations for Β§17.3(iii), Β§17.3(iii), Β§17.3 and Ch.17

These were introduced in Carlitz (1954a, 1958). The $\beta_{n}\left(x,q\right)$ are, in fact, rational functions of $q$, and not necessarily polynomials. The $A_{m,s}\left(q\right)$ are always polynomials in $q$, and the $a_{m,s}\left(q\right)$ are polynomials in $q$ for $0\leq s\leq m$.

## Β§17.3(iv) Theta Functions

See Β§Β§17.8 and 20.5.

## Β§17.3(v) Orthogonal Polynomials

See Β§Β§18.27β18.29.