products

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21—30 of 182 matching pages

21: 27.3 Multiplicative Properties

… ►

27.3.2

f(n)=\prod_{r=1}^{\nu\left(n\right)}f(p^{a_{r}}_{r}).

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20, §27.3
Permalink:: http://dlmf.nist.gov/27.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.3

\phi\left(n\right)=n\prod_{p\mathbin{|}n}(1-p^{-1}),

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
A&S Ref:: 24.3.2 I.C
Permalink:: http://dlmf.nist.gov/27.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

►

27.3.4

J_{k}\left(n\right)=n^{k}\prod_{p\mathbin{|}n}(1-p^{-k}),

ⓘ

Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $k$ : positive integer, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

►

27.3.5

d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer and $a_{r}$ : positive exponent
Permalink:: http://dlmf.nist.gov/27.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.10

f(n)=\prod_{r=1}^{\nu\left(n\right)}\left(f(p_{r})\right)^{a_{r}}.

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20
Permalink:: http://dlmf.nist.gov/27.3.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

22: 27.5 Inversion Formulas

… ►If a Dirichlet series

F(s)

generates

f(n)

, and

G(s)

generates

g(n)

, then the product

F(s)G(s)

generates ►

27.5.1

h(n)=\sum_{d\mathbin{|}n}f(d)g\left(\frac{n}{d}\right),

ⓘ

Symbols:: $d$ : positive integer, $n$ : positive integer, $f$ : function, $g$ : function and $h$ : function
Permalink:: http://dlmf.nist.gov/27.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of

f

and

g

. … ►

27.5.8

g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.

ⓘ

Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $n$ : positive integer, $f$ : function and $g$ : function
A&S Ref:: 24.3.1 II.C
Permalink:: http://dlmf.nist.gov/27.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

…

$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
$G\times H$	Cartesian product of groups $G$ and $H$ , that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$ .

24: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. … ►Their product

m

has 20 digits, twice the number of digits in the data. …

25: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The determinant of an upper or lower triangular, or diagonal, square matrix

\mathbf{A}

is the product of the diagonal elements

\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}

. … ►

1.3.14

\det\left[\frac{1}{a_{j}-b_{k}}\right]=(-1)^{n(n-1)/2}\*\prod_{1\leq j<k\leq n% }(a_{k}-a_{j})(b_{k}-b_{j})\Bigg{/}\prod^{n}_{j,k=1}(a_{j}-b_{k}).

ⓘ

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

… ►

1.3.17

\det[t_{jk}]=\prod_{1\leq j<k\leq n}(x_{j}-x_{k})\prod_{2\leq j\leq k\leq n}(b% _{j}-a_{k}).

ⓘ

Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $t_{\NVar{j}\NVar{k}}$ : elements
Referenced by:: §1.3(ii)
Permalink:: http://dlmf.nist.gov/1.3.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

… ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. … ►

1.3.20

\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},

c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $n$ : nonnegative integer
Referenced by:: Erratum (V1.2.0) Section 1.3
Permalink:: http://dlmf.nist.gov/1.3.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

…

26: 16.18 Special Cases

… ►

16.18.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.18 and Ch.16

…

27: 24.19 Methods of Computation

… ►

24.19.1

N_{2n}=\frac{2(2n)!}{(2\pi)^{2n}}\left(\prod_{p-1\mathbin{|}2n}p\right)\left(% \prod_{p}\frac{p^{2n}}{p^{2n}-1}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : integer and $p$ : prime
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(i), §24.19 and Ch.24

►

D_{2n}=\prod_{p-1\mathbin{|}2n}p,

… ►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

. …

28: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

…

29: 25.15 Dirichlet $L$ -functions

… ► ►

25.15.2

L\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{s}}\right)^{-1},

\Re s>1

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $\Re$ : real part, $p$ : prime number, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Keywords:: infinite product, product representation
Source:: Apostol (1976, p. 231)
Permalink:: http://dlmf.nist.gov/25.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.15(i), §25.15 and Ch.25

►with the product taken over all primes

p

, beginning with

p=2

. … ►

25.15.3

L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}% \right),

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $k$ : nonnegative integer, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Keywords:: finite sum
Source:: Apostol (1976, (8), p. 255)
Notes:: In the original source the upper-index of the finite sum is $k$ . We use $k-1$ since $\chi(k)=0$ .
Referenced by:: (25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.15(i), §25.15 and Ch.25

►

25.15.4

L\left(s,\chi\right)=L\left(s,\chi_{0}\right)\prod_{p\mathbin{|}k}\left(1-% \frac{\chi_{0}(p)}{p^{s}}\right),

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $k$ : nonnegative integer, $p$ : prime number, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Keywords:: product representation
Source:: Apostol (1976, p. 262)
Referenced by:: §25.15(i)
Permalink:: http://dlmf.nist.gov/25.15.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.15(i), §25.15 and Ch.25

…

30: 31.16 Mathematical Applications

… ►

§31.16(ii) Heun Polynomial Products

►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: … ►

products

21—30 of 182 matching pages

21: 27.3 Multiplicative Properties

22: 27.5 Inversion Formulas

23: 23.1 Special Notation

24: 27.15 Chinese Remainder Theorem

25: 1.3 Determinants, Linear Operators, and Spectral Expansions

26: 16.18 Special Cases

27: 24.19 Methods of Computation

28: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

29: 25.15 Dirichlet L -functions

30: 31.16 Mathematical Applications

§31.16(ii) Heun Polynomial Products

29: 25.15 Dirichlet $L$ -functions