multiplicative functions

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1: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and ►

27.3.1

f(mn)=f(m)f(n),

\left(m,n\right)=1

ⓘ

Symbols:: $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $m$ : positive integer, $n$ : positive integer and $f(n)$ : multiplicative function
Permalink:: http://dlmf.nist.gov/27.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.8

\phi\left(m\right)\phi\left(n\right)=\phi\left(mn\right)\phi\left(\left(m,n% \right)\right)/\left(m,n\right).

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

►A function

f

is completely multiplicative if

f(1)=1

and …

2: 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). …

3: 27.4 Euler Products and Dirichlet Series

… ►Every multiplicative

f

satisfies the identity … ►

27.4.2

\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.

ⓘ

Symbols:: $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers and $f$ : multiplicative function
Permalink:: http://dlmf.nist.gov/27.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.4 and Ch.27

►Euler products are used to find series that generate many functions of multiplicative number theory. The completely multiplicative function

f(n)=n^{-s}

gives the Euler product representation of the Riemann zeta function

\zeta\left(s\right)

(§25.2(i)): … ►

27.4.4

F(s)=\sum_{n=1}^{\infty}f(n)n^{-s},

ⓘ

Symbols:: $n$ : positive integer, $f$ : multiplicative function and $F(s)$ : generating function
Permalink:: http://dlmf.nist.gov/27.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §27.4 and Ch.27

…

4: 5.5 Functional Relations

… ►

§5.5(iii) Multiplication

… ►

Gauss’s Multiplication Formula

… ►

5.5.7

\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

…

5: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

17.8.2

{{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}},

\left|b/a\right|<\left|z\right|<1

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left|b/a\right|<\left|z\right|<1$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.4

{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},

\left|qa\right|<\left|bc\right|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left|qa\right|<\left|bc\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.5

{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},

\left|q\right|<\left|bcd\right|

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left|q\right|<\left|bcd\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.6

{{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},

\left|qa^{\frac{3}{2}}\right|<\left|bcd\right|

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.1) for Equation (17.8.6)
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.2.1):: The constraint $\left|qa^{\frac{3}{2}}\right|<\left|bcd\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.8

{{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),

\left|cq^{2}\right|<\left|b^{2}\right|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Source:: Verma and Jain (1983, (3.9))
Referenced by:: §17.6(i), §17.8, Erratum (V1.1.0) for Additions, Erratum (V1.2.1) for Equation (17.8.8)
Permalink:: http://dlmf.nist.gov/17.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.8, §17.8 and Ch.17

…

6: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

… ►

17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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

…

7: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

17.7.1

{{}_{2}\phi_{2}}\left({a,q/a\atop-q,b};q,-b\right)=\frac{\left(ab,bq/a;q^{2}% \right)_{\infty}}{\left(b;q\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.1.10) for Equation (17.7.1)
Permalink:: http://dlmf.nist.gov/17.7.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.10):: The constraint $|b|<1$ is not necessary and was removed.
See also:: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

… ►

17.7.2

{{}_{2}\phi_{2}}\left({a^{2},b^{2}\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}}};% q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}\right)_{\infty}}{\left(q,a^{2}b^{2% }q;q^{2}\right)_{\infty}}.

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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

… ►

17.7.4

{{}_{3}\phi_{2}}\left({a,b,q^{-n}\atop c,abq^{1-n}/c};q,q\right)=\frac{\left(c% /a,c/b;q\right)_{n}}{\left(c,c/(ab);q\right)_{n}}.

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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

… ►

17.7.12

{{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n},q^{-2n}\atop b,bq,a^{2}q^{2}};q^{2},q^% {2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}}{\left(-aq,b;q\right)_{n}}.

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

… ►

17.7.13

{{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n-2},q^{-2n}\atop b,bq,a^{2}};q^{2},q^{2}% \right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}(1-bq^{n-1})}{\left(-a,b;q\right)_% {n}(1-bq^{2n-1})}.

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

…

8: 10.44 Sums

… ►

§10.44(i) Multiplication Theorem

…

9: 10.23 Sums

… ►

§10.23(i) Multiplication Theorem

…

10: 17.6 ${{}_{2}\phi_{1}}$ Function

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17.6.1

{{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}},

|c|<|ab|

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $|c|<|ab|$ was added.
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

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17.6.5

{{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},

|b|>|q|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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

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17.6.6

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,az;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/b,z\atop az};q,b% \right),

|z|<1,|b|<1

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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17.6.7

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/b,bz;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({\ifrac{abz}{c},b% \atop bz};q,c/b\right),

|z|<1,|c|<|b|

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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17.6.10

(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}},

|z|<1

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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