17 q-Hypergeometric and Related FunctionsProperties17.2 Calculus17.4 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

Contents

1. §17.3(i) Elementary Functions
2. §17.3(ii) Gamma and Beta Functions
3. §17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers
4. §17.3(iv) Theta Functions
5. §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		$$e_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{x}^n}{{(q;q)}_n}=\frac{1}{{((1-q)x;q)}_{\mathrm{\infty }}},$$
ⓘ Defines: $e_q\left(x\right)$: $q$-exponential function Symbols: ${(a;q)}_n$: $q$-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		$$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$
ⓘ Defines: $E_q\left(x\right)$: $q$-exponential function Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-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		$${\sin }_q\left(x\right)=\frac{1}{2\mathrm{i}}(e_q\left(\mathrm{i}x\right)-e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}},$$
ⓘ Defines: ${\sin }_q\left(x\right)$: $q$-sine function Symbols: $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-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		$${\mathrm{Sin}}_q\left(x\right)=\frac{1}{2\mathrm{i}}(E_q\left(\mathrm{i}x\right)-E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{q}^{n(2n+1)}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}}.$$
ⓘ Defines: ${\mathrm{Sin}}_q\left(x\right)$: $q$-sine function Symbols: $\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-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		$${\cos }_q\left(x\right)=\frac{1}{2}(e_q\left(\mathrm{i}x\right)+e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}},$$
ⓘ Defines: ${\cos }_q\left(x\right)$: $q$-cosine function Symbols: $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-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		$${\mathrm{Cos}}_q\left(x\right)=\frac{1}{2}(E_q\left(\mathrm{i}x\right)+E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{q}^{n(2n-1)}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}}.$$
ⓘ Defines: ${\mathrm{Cos}}_q\left(x\right)$: $q$-cosine function Symbols: $\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-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

See also Suslov (2003).

§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		$${\beta }_n(x,q)={(1-q)}^{1-n}\sum _{r=0}^n{(-1)}^r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)\frac{r+1}{(1-q^{r+1})}{q}^{rx}.$$
ⓘ Defines: ${\beta }_n(x,q)$: $q$-Bernoulli polynomial Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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		$$A_{m,s}\left(q\right)=q^{\left(\genfrac{}{}{0.0pt}{}{s-m}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{m+1}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$
ⓘ Defines: $A_{m,s}\left(q\right)$: $q$-Euler number Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-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		$$a_{m,s}\left(q\right)=\frac{q^{-\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}{(1-q)}^s}{{(q;q)}_s}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{s}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$
ⓘ Defines: $a_{m,s}\left(q\right)$: $q$-Stirling number Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-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 ${\beta }_n(x,q)$ 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 $0\le s\le m$.

§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.

17.2 Calculus17.4 Basic Hypergeometric Functions

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