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

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

-Exponential Functions

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

17.3.1 $e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$

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Defines:: $e_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : 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 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$

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Defines:: $E_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : 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

-Sine Functions

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

17.3.3 $\operatorname{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left% (q;q\right)_{2n+1}},$

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Defines:: $\operatorname{sin}_{\NVar{q}}\left(\NVar{x}\right)$ : -sine function
Symbols:: $\mathrm{i}$ : imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), $e_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function, : complex base, : nonnegative integer and : 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 $\operatorname{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2% n+1}}{\left(q;q\right)_{2n+1}}.$

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Defines:: $\operatorname{Sin}_{\NVar{q}}\left(\NVar{x}\right)$ : -sine function
Symbols:: $\mathrm{i}$ : imaginary unit, $E_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : 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

-Cosine Functions

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

17.3.5 $\operatorname{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}\left(ix\right)+e_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q% \right)_{2n}},$

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Defines:: $\operatorname{cos}_{\NVar{q}}\left(\NVar{x}\right)$ : -cosine function
Symbols:: $\mathrm{i}$ : imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), $e_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function, : complex base, : nonnegative integer and : 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 $\operatorname{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}% }{\left(q;q\right)_{2n}}.$

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Defines:: $\operatorname{Cos}_{\NVar{q}}\left(\NVar{x}\right)$ : -cosine function
Symbols:: $\mathrm{i}$ : imaginary unit, $E_{\NVar{q}}\left(\NVar{x}\right)$ : -exponential function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : 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

-Bernoulli Polynomials

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

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

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Defines:: $\beta_{\NVar{n}}\left(\NVar{x},\NVar{q}\right)$ : -Bernoulli polynomial
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : complex base, : nonnegative integer, : nonnegative integer and : 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

-Euler Numbers

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

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

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Defines:: $A_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$ : -Euler number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer, : nonnegative integer and : 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

-Stirling Numbers

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

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

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Defines:: $a_{\NVar{m},\NVar{s}}\left(\NVar{q}\right)$ : -Stirling number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : -binomial coefficient (or Gaussian polynomial), : complex base, : nonnegative integer, : nonnegative integer and : 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 , and not necessarily polynomials. The $A_{m,s}\left(q\right)$ are always polynomials in , and the $a_{m,s}\left(q\right)$ are polynomials in for $0\leq s\leq m$ .