17 q-Hypergeometric and Related Functions Properties 17.2 Calculus 17.4 Basic Hypergeometric Functions

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

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Permalink:: http://dlmf.nist.gov/17.3
See also:: Annotations for Ch.17

§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

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Keywords:: $q$ -elementary functions
Permalink:: http://dlmf.nist.gov/17.3.i
See also:: Annotations for §17.3 and Ch.17

$q$ -Exponential Functions

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Keywords:: $q$ -exponential function
See also:: Annotations for §17.3(i), §17.3 and Ch.17

17.3.1		$e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$
ⓘ 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: §17.5 Permalink: http://dlmf.nist.gov/17.3.E1 Encodings: TeX, pMML, png See also: Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

17.3.2		$E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$
ⓘ 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: §17.5 Permalink: http://dlmf.nist.gov/17.3.E2 Encodings: TeX, pMML, png See also: Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

$q$ -Sine Functions

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Keywords:: $q$ -sine function
See also:: Annotations for §17.3(i), §17.3 and Ch.17

17.3.3		$\mathrm{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left(q;q% \right)_{2n+1}},$
ⓘ Defines: $\mathrm{sin}_{\NVar{q}}\left(\NVar{x}\right)$ : $q$ -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 §17.3(i), §17.3(i), §17.3 and Ch.17

17.3.4		$\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{% \left(q;q\right)_{2n+1}}.$
ⓘ Defines: $\mathrm{Sin}_{\NVar{q}}\left(\NVar{x}\right)$ : $q$ -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 §17.3(i), §17.3(i), §17.3 and Ch.17

$q$ -Cosine Functions

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Keywords:: $q$ -cosine function
See also:: Annotations for §17.3(i), §17.3 and Ch.17

17.3.5		$\mathrm{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}\left(ix\right)+e_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q\right)_{% 2n}},$
ⓘ Defines: $\mathrm{cos}_{\NVar{q}}\left(\NVar{x}\right)$ : $q$ -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 §17.3(i), §17.3(i), §17.3 and Ch.17

17.3.6		$\mathrm{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}}{\left(q% ;q\right)_{2n}}.$
ⓘ Defines: $\mathrm{Cos}_{\NVar{q}}\left(\NVar{x}\right)$ : $q$ -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 §17.3(i), §17.3(i), §17.3 and Ch.17

§17.3(ii) Gamma and Beta Functions

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Permalink:: http://dlmf.nist.gov/17.3.ii
See also:: Annotations for §17.3 and Ch.17

See §5.18.

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

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Permalink:: http://dlmf.nist.gov/17.3.iii
See also:: Annotations for §17.3 and Ch.17

$q$ -Bernoulli Polynomials

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Keywords:: $q$ -Bernoulli polynomials
See also:: Annotations for §17.3(iii), §17.3 and Ch.17

17.3.7		$\beta_{n}\left(x,q\right)=(1-q)^{1-n}\sum_{r=0}^{n}(-1)^{r}\genfrac{(}{)}{0.0% pt}{}{n}{r}\frac{r+1}{(1-q^{r+1})}q^{rx}.$
ⓘ 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 §17.3(iii), §17.3(iii), §17.3 and Ch.17

$q$ -Euler Numbers

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Keywords:: $q$ -Euler numbers
See also:: Annotations for §17.3(iii), §17.3 and Ch.17

17.3.8		$A_{m,s}\left(q\right)=q^{\genfrac{(}{)}{0.0pt}{}{s-m}{2}+\genfrac{(}{)}{0.0pt}% {}{s}{2}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}% {0.0pt}{}{m+1}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$
ⓘ 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 §17.3(iii), §17.3(iii), §17.3 and Ch.17

$q$ -Stirling Numbers

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Keywords:: $q$ -Stirling numbers
See also:: Annotations for §17.3(iii), §17.3 and Ch.17

17.3.9		$a_{m,s}\left(q\right)=\frac{q^{-\genfrac{(}{)}{0.0pt}{}{s}{2}}(1-q)^{s}}{\left% (q;q\right)_{s}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}% \genfrac{[}{]}{0.0pt}{}{s}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$
ⓘ 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 §17.3(iii), §17.3(iii), §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 $0\leq s\leq m$ .

§17.3(iv) Theta Functions

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Permalink:: http://dlmf.nist.gov/17.3.iv
See also:: Annotations for §17.3 and Ch.17

See §§17.8 and 20.5.

§17.3(v) Orthogonal Polynomials

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Permalink:: http://dlmf.nist.gov/17.3.v
See also:: Annotations for §17.3 and Ch.17

See §§18.27–18.29.

§17.3 q-Elementary and q-Special Functions

Contents

§17.3(i) Elementary Functions

q-Exponential Functions

q-Sine Functions

q-Cosine Functions