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

## §17.3(i) Elementary Functions

### $q$-Exponential Functions

 17.3.1 $\mathop{e_{q}\/}\nolimits\!\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^% {n}}{\left(q;q\right)_{n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$ Defines: $\mathop{e_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-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: §17.5 Permalink: http://dlmf.nist.gov/17.3.E1 Encodings: TeX, pMML, png See also: Annotations for 17.3(i)
 17.3.2 $\mathop{E_{q}\/}\nolimits\!\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^% {\binom{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$ Defines: $\mathop{E_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-exponential function Symbols: $\binom{\NVar{m}}{\NVar{n}}$: 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: §17.5 Permalink: http://dlmf.nist.gov/17.3.E2 Encodings: TeX, pMML, png See also: Annotations for 17.3(i)

### $q$-Sine Functions

 17.3.3 $\mathop{\mathrm{sin}_{q}\/}\nolimits\!\left(x\right)=\frac{1}{2i}(\mathop{e_{q% }\/}\nolimits\!\left(ix\right)-\mathop{e_{q}\/}\nolimits\!\left(-ix\right))=% \sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left(q;q\right)_{2n+1}},$ Defines: $\mathop{\mathrm{sin}_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-sine function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\mathop{e_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-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 17.3(i)
 17.3.4 $\mathop{\mathrm{Sin}_{q}\/}\nolimits\!\left(x\right)=\frac{1}{2i}(\mathop{E_{q% }\/}\nolimits\!\left(ix\right)-\mathop{E_{q}\/}\nolimits\!\left(-ix\right))=% \sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{\left(q;q% \right)_{2n+1}}.$ Defines: $\mathop{\mathrm{Sin}_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-sine function Symbols: $\mathop{E_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-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 17.3(i)

### $q$-Cosine Functions

 17.3.5 $\mathop{\mathrm{cos}_{q}\/}\nolimits\!\left(x\right)=\frac{1}{2}(\mathop{e_{q}% \/}\nolimits\!\left(ix\right)+\mathop{e_{q}\/}\nolimits\!\left(-ix\right))=% \sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q\right)_{2n}},$ Defines: $\mathop{\mathrm{cos}_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-cosine function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\mathop{e_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-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 17.3(i)
 17.3.6 $\mathop{\mathrm{Cos}_{q}\/}\nolimits\!\left(x\right)=\frac{1}{2}(\mathop{E_{q}% \/}\nolimits\!\left(ix\right)+\mathop{E_{q}\/}\nolimits\!\left(-ix\right))=% \sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}}{\left(q;q\right)% _{2n}}.$ Defines: $\mathop{\mathrm{Cos}_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-cosine function Symbols: $\mathop{E_{\NVar{q}}\/}\nolimits\!\left(\NVar{x}\right)$: $q$-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 17.3(i)

See §5.18.

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

### $q$-Bernoulli Polynomials

 17.3.7 $\mathop{\beta_{n}\/}\nolimits\!\left(x,q\right)=(1-q)^{1-n}\sum_{r=0}^{n}(-1)^% {r}\binom{n}{r}\frac{r+1}{(1-q^{r+1})}q^{rx}.$ Defines: $\mathop{\beta_{\NVar{n}}\/}\nolimits\!\left(\NVar{x},\NVar{q}\right)$: $q$-Bernoulli polynomial Symbols: $\binom{\NVar{m}}{\NVar{n}}$: 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 17.3(iii)

### $q$-Euler Numbers

 17.3.8 $\mathop{A_{m,s}\/}\nolimits\!\left(q\right)=q^{\binom{s-m}{2}+\binom{s}{2}}% \sum_{j=0}^{s}(-1)^{j}q^{\binom{j}{2}}\genfrac{[}{]}{0.0pt}{}{m+1}{j}_{q}\frac% {(1-q^{s-j})^{m}}{(1-q)^{m}}.$ Defines: $\mathop{A_{\NVar{m},\NVar{s}}\/}\nolimits\!\left(\NVar{q}\right)$: $q$-Euler number Symbols: $\binom{\NVar{m}}{\NVar{n}}$: 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 17.3(iii)

### $q$-Stirling Numbers

 17.3.9 $\mathop{a_{m,s}\/}\nolimits\!\left(q\right)=\frac{q^{-\binom{s}{2}}(1-q)^{s}}{% \left(q;q\right)_{s}}\sum_{j=0}^{s}(-1)^{j}q^{\binom{j}{2}}\genfrac{[}{]}{0.0% pt}{}{s}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$ Defines: $\mathop{a_{\NVar{m},\NVar{s}}\/}\nolimits\!\left(\NVar{q}\right)$: $q$-Stirling number Symbols: $\binom{\NVar{m}}{\NVar{n}}$: 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 17.3(iii)

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

## §17.3(iv) Theta Functions

See §§17.8 and 20.5.

## §17.3(v) Orthogonal Polynomials

See §§18.2718.29.