multiplicative functions

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11: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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17.9.3

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),

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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 and $z$ : complex variable
Referenced by:: Erratum (V1.0.23) for Equation (17.9.3)
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.23):: The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.3_5

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/a,c/b;q\right)_{% \infty}}{\left(c,c/(ab);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c% \atop qab/c,0};q,q\right)+\frac{\left(a,b,abz/c;q\right)_{\infty}}{\left(c,ab/% c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,c/b,z\atop qc/(ab),0};q,q% \right),

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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
Proof sketch:: Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\rightarrow 0$ . 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

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17.9.6

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(e/a,de/(% bc);q\right)_{\infty}}{\left(e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),

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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/17.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.7

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(b,de/(ab)% ,de/(bc);q\right)_{\infty}}{\left(d,e,de/(abc);q\right)_{\infty}}\*{{}_{3}\phi% _{2}}\left({d/b,e/b,de/(abc)\atop de/(ab),de/(bc)};q,b\right),

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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/17.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.11

{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(e/c,d/c;q% \right)_{n}}{\left(e,d;q\right)_{n}}c^{n}{{}_{3}\phi_{2}}\left({q^{-n},c,% \ifrac{cbq^{1-n}}{(de)}\atop\ifrac{cq^{1-n}}{e},\ifrac{cq^{1-n}}{d}};q,q\right),

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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
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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12: 27.5 Inversion Formulas

… ►The set of all number-theoretic functions

f

with

f(1)\neq 0

forms an abelian group under Dirichlet multiplication, with the function

\left\lfloor 1/n\right\rfloor

in (27.2.5) as identity element; see Apostol (1976, p. 129). The multiplicative functions are a subgroup of this group. …

13: 17.11 Transformations of $q$ -Appell Functions

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17.11.1

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),

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Symbols:: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$ : first $q$ -Appell 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\Phi^{(1)}$ 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

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17.11.2

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : second $q$ -Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\left(q;q\right)_{r}$ .
Correction (effective with 1.0.10):: The notation for $\Phi^{(2)}$ 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

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.

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Symbols:: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : third $q$ -Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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 $\Phi^{(3)}$ 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},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).

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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, $m$ : 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

14: 13.26 Addition and Multiplication Theorems

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§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$

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15: 13.13 Addition and Multiplication Theorems

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§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

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16: 4.35 Identities

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§4.35(iii) Multiples of the Argument

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17: 4.21 Identities

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§4.21(iii) Multiples of the Argument

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18: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of

q(z)

. …

19: 4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

► ►►

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
…

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20: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi