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

§17.3 -Elementary and -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:: -elementary functions
Permalink:: http://dlmf.nist.gov/17.3.i

¶ -Exponential Functions

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

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

¶ -Sine Functions

Keywords:: -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:: $\mathop{\mathrm{sin}_{{q}}\/}\nolimits\!\left(x\right)$ : -sine function
Symbols:: $\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$ : -exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : 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:: $\mathop{\mathrm{Sin}_{{q}}\/}\nolimits\!\left(x\right)$ : -sine function
Symbols:: $\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$ : -exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/17.3.E4
Encodings:: TeX, pMML, png

¶ -Cosine Functions

Keywords:: -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:: $\mathop{\mathrm{cos}_{{q}}\/}\nolimits\!\left(x\right)$ : -cosine function
Symbols:: $\mathop{e_{{q}}\/}\nolimits\!\left(x\right)$ : -exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : 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:: $\mathop{\mathrm{Cos}_{{q}}\/}\nolimits\!\left(x\right)$ : -cosine function
Symbols:: $\mathop{E_{{q}}\/}\nolimits\!\left(x\right)$ : -exponential function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer and : 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

¶ -Bernoulli Polynomials

Keywords:: -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:: $\mathop{\beta_{{n}}\/}\nolimits\!\left(x,q\right)$ : -Bernoulli polynomial
Symbols:: $\binom{m}{n}$ : binomial coefficient, : complex base, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/17.3.E7
Encodings:: TeX, pMML, png

¶ -Euler Numbers

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

¶ -Stirling Numbers

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