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

Permalink:: http://dlmf.nist.gov/17.3

§17.3(i) Elementary Functions
§17.3(ii) Gamma and Beta Functions
§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers
§17.3(iv) Theta Functions
§17.3(v) Orthogonal Polynomials

§17.3(i) Elementary Functions

Keywords:: $q$ -elementary functions
Permalink:: http://dlmf.nist.gov/17.3.i

¶ $q$ -Exponential Functions

Keywords:: $q$ -exponential function

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:: $\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function
Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

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:: $\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

¶ $q$ -Sine Functions

Keywords:: $q$ -sine function

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:: $\mathop{\mathrm{sin}_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -sine function
Symbols:: $\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.3.E3
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Sin}_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -sine function
Symbols:: $\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

¶ $q$ -Cosine Functions

Keywords:: $q$ -cosine function

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:: $\mathop{\mathrm{cos}_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -cosine function
Symbols:: $\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.3.E5
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Cos}_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -cosine function
Symbols:: $\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$ : $q$ -exponential function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

§17.3(ii) Gamma and Beta Functions

Permalink:: http://dlmf.nist.gov/17.3.ii

See §5.18.

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

Permalink:: http://dlmf.nist.gov/17.3.iii

¶ $q$ -Bernoulli Polynomials

Keywords:: $q$ -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:: $\mathop{\beta _{{n}}\/}\nolimits\!\left(x,q\right)$ : $q$ -Bernoulli polynomial
Symbols:: $\binom{m}{n}$ : 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

¶ $q$ -Euler Numbers

Keywords:: $q$ -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:: $\mathop{A_{{m,s}}\/}\nolimits\!\left(q\right)$ : $q$ -Euler number
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -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

¶ $q$ -Stirling Numbers

Keywords:: $q$ -Stirling numbers

17.3.9 $\mathop{a_{{m,s}}\/}\nolimits\!\left(q\right)=\frac{q^{{-\binom{s}{2}}}(1-q)^{s}}{\left(q;q\right)_{{s}}}\sum _{{j=0}}^{s}(-1)^{j}q^{{\binom{j}{2}}}\genfrac{[}{]}{0.0pt}{}{s}{j}_{{q}}\frac{(1-q^{{s-j}})^{m}}{(1-q)^{m}}.$

Defines:: $\mathop{a_{{m,s}}\/}\nolimits\!\left(q\right)$ : $q$ -Stirling number
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $j$ : nonnegative integer, $m$ : nonnegative integer and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.3.E9
Encodings:: TeX, pMML, png

These were introduced in Carlitz (1954b, 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

Permalink:: http://dlmf.nist.gov/17.3.iv

See §§17.8 and 20.5.

§17.3(v) Orthogonal Polynomials

Permalink:: http://dlmf.nist.gov/17.3.v

See §§18.27–18.29.

§17.3 -Elementary and -Special Functions

Contents

§17.3(i) Elementary Functions

¶ -Exponential Functions

¶ -Sine Functions

¶ -Cosine Functions