q-sine function

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1: 17.3 $q$-Elementary and $q$-Special Functions

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$q$-Sine Functions

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17.3.3 $${\sin }_q\left(x\right)=\frac{1}{2\mathrm{i}}(e_q\left(\mathrm{i}x\right)-e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}},$$

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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:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.4 $${\mathrm{Sin}}_q\left(x\right)=\frac{1}{2\mathrm{i}}(E_q\left(\mathrm{i}x\right)-E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{q}^{n(2n+1)}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}}.$$

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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:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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2: 28.4 Fourier Series

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28.4.3 $${\mathrm{se}}_{2n+1}(z,q)=\sum _{m=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q)\sin (2m+1)z,$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(i), §28.4 and Ch.28

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28.4.4 $${\mathrm{se}}_{2n+2}(z,q)=\sum _{m=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q)\sin (2m+2)z.$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(i), §28.4 and Ch.28

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3: 28.6 Expansions for Small $q$

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28.6.23 $${\mathrm{se}}_1(z,q)=\sin z-\frac{1}{8}q\sin 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\sin 5z+2\sin 3z-\sin z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\sin 7z+\frac{8}{9}\sin 5z-\frac{1}{3}\sin 3z-2\sin z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.25 $${\mathrm{se}}_2(z,q)=\sin 2z-\frac{1}{12}q\sin 4z+\frac{1}{128}{q}^2\left(\frac{1}{3}\sin 6z-\frac{4}{9}\sin 2z\right)+\mathrm{\cdots }.$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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4: 28.14 Fourier Series

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28.14.3 $${\mathrm{se}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q)\sin (\nu +2m)z,$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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