$g, h$	positive integers.
…
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
…
$\operatorname{diag}\mathbf{A}$	Transpose of $[A_{11},A_{22},\dots,A_{gg}]$ .
…
$\mathbf{J}_{2g}$	$\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ .
…
$S_{1}S_{2}$	set of all elements of the form “ $\mbox{element of $S_{1}$}\times\mbox{element of $S_{2}$}$ ”.
$S_{1}/S_{2}$	set of all elements of $S_{1}$ , modulo elements of $S_{2}$ . Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$ . (For an example see §20.12(ii).)
…

…

24: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

►

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

… ►where

m_{1},m_{2},\dots,m_{r}

are arbitrary nonnegative integers. …

25: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: … ►The coefficients of the power series of

a_{2n}\left(q\right)

b_{2n}\left(q\right)

and also

a_{2n+1}\left(q\right)

b_{2n+1}\left(q\right)

are the same until the terms in

q^{2n-2}

and

q^{2n}

, respectively. … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. Here

j=1

for

a_{2n}\left(q\right)

j=2

for

b_{2n+2}\left(q\right)

, and

j=3

for

a_{2n+1}\left(q\right)

and

b_{2n+1}\left(q\right)

. … ►

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

…

26: 3.1 Arithmetics and Error Measures

… ►with

b_{0}=1

and all allowable choices of

E

p

s

, and

b_{j}

. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. …The integers

p

E_{{\rm min}}

, and

E_{{\rm max}}

are characteristics of the machine. … ►The respective machine precisions are

\frac{1}{2}\epsilon_{M}=0.596\times 10^{-7}

\frac{1}{2}\epsilon_{M}=0.111\times 10^{-15}

and

\frac{1}{2}\epsilon_{M}=0.963\times 10^{-34}

. … ►

N_{{\rm min}}\leq x\leq N_{{\rm max}}

, and …

27: 17.13 Integrals

… ►In this section, for the function

\Gamma_{q}

see §5.18(ii). ►

17.13.1

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}\,{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},

ⓘ

Symbols:: $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.2

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.3

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $q$ : complex base
Referenced by:: Erratum (V1.0.1) for Equation (17.13.3)
Permalink:: http://dlmf.nist.gov/17.13.E3
Encodings:: pMML, png, TeX
Errata (effective with 1.0.1):: Originally the differential was identified incorrectly as a $q$ -differential; the correct differential is $\,\mathrm{d}t$ .
See also:: Annotations for §17.13, §17.13 and Ch.17

►

17.13.4

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13, §17.13 and Ch.17

…

28: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

m=1(1)100

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

\sum_{k=0}^{\infty}(2k+1)^{-n}

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

n=1,2,\dotsc

, 18D. ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

29: 17.14 Constant Term Identities

… ►

17.14.1

\frac{\left(q;q\right)_{a_{1}+a_{2}+\cdots+a_{n}}}{\left(q;q\right)_{a_{1}}% \left(q;q\right)_{a_{2}}\cdots\left(q;q\right)_{a_{n}}}=\mbox{ coeff. of }x_{1% }^{0}x_{2}^{0}\cdots x_{n}^{0}\mbox{ in }\prod_{1\leq j<k\leq n}\left(\frac{x_% {j}}{x_{k}};q\right)_{a_{j}}\left(\frac{qx_{k}}{x_{j}};q\right)_{a_{k}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

… ►

17.14.2

\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1}q^{2};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z% ^{-1}q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ % coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q% ;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1}q;q\right% )_{\infty}}=\frac{H(q)}{\left(-q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.3

\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z^{-1}% q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ coeff.% of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1};q\right)_{% \infty}}=\frac{G(q)}{\left(-q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.4

\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{2}% \right)_{n}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{% \infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{% \left(-z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(z^{-1};q% ^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}\mbox{ coeff. of % }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-2};q^{4}\right)_{% \infty}}=\frac{G(q^{4})}{\left(q;q^{2}\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

17.14.5

\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{% 2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right% )_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty% }}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(% z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}% \mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-% z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(q^{4}z^{% -2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{\left(q;q^{2}\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

…

30: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. …For example, the tangent numbers

T_{n}

can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. … ►If

\widetilde{N}_{2n}

denotes the right-hand side of (24.19.1) but with the second product taken only for

p\leq\left\lfloor(\pi e)^{-1}2n\right\rfloor+1

, then

N_{2n}=\left\lceil\widetilde{N}_{2n}\right\rceil

for

n\geq 2

. … ►For algorithms for computing

B_{n}

E_{n}

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

…

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21—30 of 786 matching pages

21: 16.4 Argument Unity

22: 24.2 Definitions and Generating Functions

23: 21.1 Special Notation

24: 17.7 Special Cases of Higher ϕ s r Functions

§17.7(i) ϕ 2 2 Functions

q -Analog of Bailey’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Gauss’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Dixon’s F 2 3 ⁡ ( 1 ) Sum

25: 28.6 Expansions for Small q

§28.6(ii) Functions ce n and se n

26: 3.1 Arithmetics and Error Measures

27: 17.13 Integrals

28: 24.20 Tables

29: 17.14 Constant Term Identities

30: 24.19 Methods of Computation

§24.19(ii) Values of B n Modulo p

24: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

25: 28.6 Expansions for Small $q$

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

§24.19(ii) Values of $B_{n}$ Modulo $p$