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1: 17.11 Transformations of $q$-Appell Functions

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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

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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

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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

2: 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 …

3: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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17.7.2 $${}_2{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a^2,b^2}{ab{q}^{\frac{1}{2}},-ab{q}^{\frac{1}{2}}};q,-q)=\frac{{(a^{2}q,b^{2}q;q^2)}_{\mathrm{\infty }}}{{(q,a^2{b}^{2}q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

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/17.7.E2

Encodings:

pMML, png, TeX

See also:

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

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17.7.4 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,q^{-n}}{c,ab{q}^{1-n}/c};q,q)=\frac{{(c/a,c/b;q)}_n}{{(c,c/(ab);q)}_n}.$$

ⓘ

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.E4

Encodings:

pMML, png, TeX

See also:

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

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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

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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

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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

…

4: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

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18.29.1 $${(bc,bd,cd;q)}_n\left(Q_n({\mathrm{e}}^{\mathrm{i}\theta };a,b,c,d\mid q)+Q_n({\mathrm{e}}^{-\mathrm{i}\theta };a,b,c,d\mid q)\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.29 and Ch.18

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18.29.2 $$Q_n(z;a,b,c,d\mid q)\sim \frac{z^n{(a{z}^{-1},b{z}^{-1},c{z}^{-1},d{z}^{-1};q)}_{\mathrm{\infty }}}{{(z^{-2},bc,bd,cd;q)}_{\mathrm{\infty }}},$$ $n\to \mathrm{\infty }$; $z,a,b,c,d,q$ fixed.

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/18.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.29 and Ch.18

…

5: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

ⓘ

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/17.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.2 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,bz,cq/(abz),dq/(abz);q)}_{\mathrm{\infty }}}{{(q/a,q/b,c,d;q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{abz/c,abz/d}{az,bz};q,\frac{cd}{abz}).$$

ⓘ

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/17.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

ⓘ

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 $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.4 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{e,f}{aq/c,aq/d};q,\frac{aq}{ef})=\frac{{(q/c,q/d,aq/e,aq/f;q)}_{\mathrm{\infty }}}{{(aq,q/a,aq/(cd),aq/(ef);q)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(1-a{q}^{2n}){(c,d,e,f;q)}_n}{(1-a){(aq/c,aq/d,aq/e,aq/f;q)}_n}{\left(\frac{q{a}^3}{cdef}\right)}^n{q}^{n^2}.$$

ⓘ

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 $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.5 $$\frac{{(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)}_{\mathrm{\infty }}}{{(fa,ga,f/a,g/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^2}{bcdefg})=\frac{{(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)}_{\mathrm{\infty }}}{{(fa,q/(fa),aq/f,f/a,g/f,fg,q{f}^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{f^2,qf,-qf,fb,fc,fd,fe,fg}{f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^2}{bcdefg})+\mathrm{idem}(f;g).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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/17.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

…

6: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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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

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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

Referenced by:

Erratum (V1.2.1) for Equation (17.8.2)

Permalink:

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

Encodings:

pMML, png, TeX

Correction (effective with 1.2.1):

The constraint was added.

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

Referenced by:

Erratum (V1.2.1) for Equation (17.8.4)

Permalink:

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

Encodings:

pMML, png, TeX

Correction (effective with 1.2.1):

The constraint was added.

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

Referenced by:

Erratum (V1.2.1) for Equation (17.8.5)

Permalink:

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

Encodings:

pMML, png, TeX

Correction (effective with 1.2.1):

The constraint was added.

See also:

Annotations for §17.8, §17.8 and Ch.17

…

7: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

… ►

17.9.3_5 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\frac{{(c/a,c/b;q)}_{\mathrm{\infty }}}{{(c,c/(ab);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,abz/c}{qab/c,0};q,q)+\frac{{(a,b,abz/c;q)}_{\mathrm{\infty }}}{{(c,ab/c,z;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,c/b,z}{qc/(ab),0};q,q),$$

ⓘ

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 $z$: complex variable

Proof sketch:

Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\to 0$. Then replace $a$, $b$, $c$, $e$ with $abz/c$, $a$, $b$, $c$ respectively.

Referenced by:

§17.9(i), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/17.9.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

… ►

17.9.6 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,de/(abc))=\frac{{(e/a,de/(bc);q)}_{\mathrm{\infty }}}{{(e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,d/b,d/c}{d,de/(bc)};q,e/a),$$

ⓘ

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/17.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

►

17.9.7 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,de/(abc))=\frac{{(b,de/(ab),de/(bc);q)}_{\mathrm{\infty }}}{{(d,e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{d/b,e/b,de/(abc)}{de/(ab),de/(bc)};q,b),$$

ⓘ

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/17.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

… ►

17.9.13 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,\frac{de}{abc})=\frac{{(e/b,e/c;q)}_{\mathrm{\infty }}}{{(e,e/(bc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{d/a,b,c}{d,bcq/e};q,q)+\frac{{(d/a,b,c,de/(bc);q)}_{\mathrm{\infty }}}{{(d,e,bc/e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{e/b,e/c,de/(abc)}{de/(bc),eq/(bc)};q,q).$$

ⓘ

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

Referenced by:

(17.9.3_5)

Permalink:

http://dlmf.nist.gov/17.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

… ►

17.9.14 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,b,c}{d,e,f};q,q)=\frac{{(e/a,f/a;q)}_n}{{(e,f;q)}_n}{a}^n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,d/b,d/c}{d,a{q}^{1-n}/e,a{q}^{1-n}/f};q,q)=\frac{{(a,ef/(ab),ef/(ac);q)}_n}{{(e,f,ef/(abc);q)}_n}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},e/a,f/a,ef/(abc)}{ef/(ab),ef/(ac),q^{1-n}/a};q,q).$$

ⓘ

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/17.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

…

8: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). …

9: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►

17.6.1 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,c/(ab))=\frac{{(c/a,c/b;q)}_{\mathrm{\infty }}}{{(c,c/(ab);q)}_{\mathrm{\infty }}},$$ .

ⓘ

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

Referenced by:

Erratum (V1.2.0) for Equation (17.6.1)

Permalink:

http://dlmf.nist.gov/17.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This constraint was added.

See also:

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

… ►

17.6.5 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{aq/b};q,-q/b)=\frac{{(-q;q)}_{\mathrm{\infty }}{(aq,a{q}^2/b^2;q^2)}_{\mathrm{\infty }}}{{(-q/b,aq/b;q)}_{\mathrm{\infty }}},$$ $|b|>|q|$.

ⓘ

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 and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.6.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

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. …

10: 17.4 Basic Hypergeometric Functions

… ►

17.4.3 $${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}=\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(q/b_1,q/b_2,\mathrm{\dots },q/b_s;q)}_n}{{(q/a_1,q/a_2,\mathrm{\dots },q/a_r;q)}_n}{\left(\frac{b_1{b}_2\mathrm{\cdots }{b}_s}{a_1{a}_2\mathrm{\cdots }{a}_{r}z}\right)}^n.$$

ⓘ

Defines:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§17.8

Permalink:

http://dlmf.nist.gov/17.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

… ►

17.4.6 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n},$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function

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, $m$: nonnegative integer, $n$: 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 (17.4.6)

Permalink:

http://dlmf.nist.gov/17.4.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.3):

The explicit usage of ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

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 §17.4(iii), §17.4 and Ch.17

►

17.4.7 $${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)=\sum _{m,n\ge 0}\frac{{(a,b;q)}_m{(a^{\prime },b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}},$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function

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, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E7

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 §17.4(iii), §17.4 and Ch.17

►

17.4.8 $${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)=\sum _{m,n\ge 0}\frac{{(a,b;q)}_{m+n}{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n}.$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$: fourth $q$-Appell function

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, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(4)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

…