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1: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

… ►

17.7.6 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-2n},b,c}{q^{1-2n}/b,q^{1-2n}/c};q,\frac{q^{2-n}}{bc})=\frac{{(b,c;q)}_n{(q,bc;q)}_{2n}}{{(q,bc;q)}_n{(b,c;q)}_{2n}}.$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

►

Continued Fractions

►For continued-fraction representations of a ratio of ${}_3{}^{}\varphi _2^{}$ functions, see Cuyt et al. (2008, pp. 399–400). … ►

17.7.21 $$\sum _{k=0}^n\frac{(1-a{p}^k{q}^k)(1-b{p}^k{q}^{-k})}{(1-a)(1-b)}\frac{{(a,b;p)}_k{(c,a/(bc);q)}_k}{{(q,aq/b;q)}_k{(ap/c,bcp;p)}_k}{q}^k=\frac{{(ap,bp;p)}_n{(cq,aq/(bc);q)}_n}{{(q,aq/b;q)}_n{(ap/c,bcp;p)}_n},$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

►

17.7.22 $$\sum _{k=-m}^n\frac{(1-ad{p}^k{q}^k)(1-b{p}^k/(d{q}^k))}{(1-ad)(1-(b/d))}\frac{{(a,b;p)}_k{(c,a{d}^2/(bc);q)}_k}{{(dq,adq/b;q)}_k{(adp/c,bcp/d;p)}_k}{q}^k=\frac{(1-a)(1-b)(1-c)(1-(a{d}^2/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{{(ap,bp;p)}_n{(cq,a{d}^{2}q/(bc);q)}_n}{{(dq,adq/b;q)}_n{(adp/c,bcp/d;p)}_n}-\frac{{(c/(ad),d/(bc);p)}_{m+1}{(1/d,b/(ad);q)}_{m+1}}{{(1/c,bc/(a{d}^2);q)}_{m+1}{(1/a,1/b;p)}_{m+1}}\right),$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer, $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

…

2: 12.6 Continued Fraction

§12.6 Continued Fraction

►For a continued-fraction expansion of the ratio $U(a,x)/U(a-1,x)$ see Cuyt et al. (2008, pp. 340–341).

3: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►

17.6.13 $${}_2{}^{}\varphi _1^{}(a,b;c;q,q)+\frac{{(q/c,a,b;q)}_{\mathrm{\infty }}}{{(c/q,aq/c,bq/c;q)}_{\mathrm{\infty }}}{}_2{}^{}\varphi _1^{}(aq/c,bq/c;q^2/c;q,q)=\frac{{(q/c,abq/c;q)}_{\mathrm{\infty }}}{{(aq/c,bq/c;q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

… ►

17.6.29 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\left(\frac{-1}{2\pi \mathrm{i}}\right)\frac{{(a,b;q)}_{\mathrm{\infty }}}{{(q,c;q)}_{\mathrm{\infty }}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{{(q^{1+\zeta },c{q}^{\zeta };q)}_{\mathrm{\infty }}}{{(a{q}^{\zeta },b{q}^{\zeta };q)}_{\mathrm{\infty }}}\frac{\pi {(-z)}^{\zeta }}{\sin \left(\pi \zeta \right)}d\zeta ,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\sin z$: sine function, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.6.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

►where , , and the contour of integration separates the poles of ${(q^{1+\zeta },c{q}^{\zeta };q)}_{\mathrm{\infty }}/\sin \left(\pi \zeta \right)$ from those of $1/{(a{q}^{\zeta },b{q}^{\zeta };q)}_{\mathrm{\infty }}$, and the infimum of the distances of the poles from the contour is positive. ►

§17.6(vi) Continued Fractions

►For continued-fraction representations of the ${}_2{}^{}\varphi _1^{}$ function, see Cuyt et al. (2008, pp. 395–399).

4: 27.19 Methods of Computation: Factorization

… ►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …

5: 17.2 Calculus

… ► … ►

17.2.5 $${(a_1,a_2,\mathrm{\dots },a_r;q)}_n=\prod _{j=1}^r{(a_j;q)}_n,$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $r$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

►

17.2.6 $${(a_1,a_2,\mathrm{\dots },a_r;q)}_{\mathrm{\infty }}=\prod _{j=1}^r{(a_j;q)}_{\mathrm{\infty }}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $j$: nonnegative integer and $r$: nonnegative integer

Referenced by:

§17.2(i)

Permalink:

http://dlmf.nist.gov/17.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23)

Permalink:

http://dlmf.nist.gov/17.2.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$.

Reported 2017-06-26 by Jason Zhao

See also:

Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.23 $$\frac{{(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}{{(a^{\frac{1}{k}},{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}=\frac{{(a{q}^k;q^k)}_n}{{(a;q^k)}_n}=\frac{1-a{q}^{kn}}{1-a},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23)

Permalink:

http://dlmf.nist.gov/17.2.E23

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$ instead of ${(a{q}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$.

Reported 2017-06-26 by Jason Zhao

See also:

Annotations for §17.2(i), §17.2 and Ch.17

…

6: 10.55 Continued Fractions

§10.55 Continued Fractions

►For continued fractions for ${𝗃}_{n+1}\left(z\right)/{𝗃}_n\left(z\right)$ and ${𝗂}_{n+1}^{(1)}\left(z\right)/{𝗂}_n^{(1)}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).

7: 3.10 Continued Fractions

§3.10 Continued Fractions

… ►

Stieltjes Fractions

… ►is called a Stieltjes fraction ($S$-fraction). … ►

Jacobi Fractions

… ►The continued fraction …

8: 17.11 Transformations of $q$-Appell Functions

… ►

17.11.1 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\frac{{(a,bx,b^{\prime }y;q)}_{\mathrm{\infty }}}{{(c,x,y;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,x,y}{bx,b^{\prime }y};q,a),$$

ⓘ

Symbols:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

►

17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)

Permalink:

http://dlmf.nist.gov/17.11.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

►

17.11.3 $${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)=\frac{{(a,bx;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a^{\prime },b^{\prime };q)}_n{(x;q)}_r{(c/a;q)}_{n+r}{a}^r{y}^n}{{(q,c/a;q)}_n{(q,bx;q)}_r}.$$

ⓘ

Symbols:

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

… ►

17.11.4 $$\sum _{m_1,\mathrm{\dots },m_n\geqq 0}\frac{{(a;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}{(b_1;q)}_{m_1}{(b_2;q)}_{m_2}\mathrm{\cdots }{(b_n;q)}_{m_n}{x}_1^{m_1}{x}_2^{m_2}\mathrm{\cdots }{x}_n^{m_n}}{{(q;q)}_{m_1}{(q;q)}_{m_2}\mathrm{\cdots }{(q;q)}_{m_n}{(c;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}}=\frac{{(a,b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n;q)}_{\mathrm{\infty }}}{{(c,x_1,x_2,\mathrm{\dots },x_n;q)}_{\mathrm{\infty }}}{}_{n+1}{}^{}\varphi _n^{}(\genfrac{}{}{0pt}{}{c/a,x_1,x_2,\mathrm{\dots },x_n}{b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n};q,a).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.11 and Ch.17

9: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

… ►If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. …Related multiplicative properties are … ►A function $f$ is completely multiplicative if $f(1)=1$ and … ►If $f$ is completely multiplicative, then (27.3.2) becomes …

10: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

… ►

17.8.1 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-z)}^n{q}^{n(n-1)/2}={(q,z,q/z;q)}_{\mathrm{\infty }};$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.2 $${}_1{}^{}\psi _1^{}(\genfrac{}{}{0pt}{}{a}{b};q,z)=\frac{{(q,b/a,az,q/(az);q)}_{\mathrm{\infty }}}{{(b,q/a,z,b/(az);q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.3 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{n(3n-1)/2}{z}^{3n}(1+z{q}^n)={(q,-z,-q/z;q)}_{\mathrm{\infty }}{(q{z}^2,q/z^2;q^2)}_{\mathrm{\infty }}.$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.4 $${}_2{}^{}\psi _2^{}(b,c;aq/b,aq/c;q,-aq/(bc))=\frac{{(aq/(bc);q)}_{\mathrm{\infty }}{(a{q}^2/b^2,a{q}^2/c^2,q^2,aq,q/a;q^2)}_{\mathrm{\infty }}}{{(aq/b,aq/c,q/b,q/c,-aq/(bc);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

►

17.8.5 $${}_3{}^{}\psi _3^{}(\genfrac{}{}{0pt}{}{b,c,d}{q/b,q/c,q/d};q,\frac{q}{bcd})=\frac{{(q,q/(bc),q/(bd),q/(cd);q)}_{\mathrm{\infty }}}{{(q/b,q/c,q/d,q/(bcd);q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

…