positive sums

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21: 1.7 Inequalities

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§1.7(i) Finite Sums

►In this subsection $A$ and $B$ are positive constants. ►

Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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22: 27.8 Dirichlet Characters

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27.8.6

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

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23: 1.2 Elementary Algebra

… ►If $m_1,m_2,\mathrm{\dots },m_n$ are positive integers and , then there exist polynomials $f_j(x)$, , such that …

24: 27.3 Multiplicative Properties

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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27.3.7 $${\sigma }_{\alpha }\left(m\right){\sigma }_{\alpha }\left(n\right)=\sum _{d|\left(m,n\right)}{d}^{\alpha }{\sigma }_{\alpha }\left(\frac{mn}{d^2}\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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25: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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${\sum }_{\left(m,n\right)=1}$	sum taken over $m$, $1\le m\le n$ and $m$ relatively prime to $n$.
$p,p_1,p_2,\mathrm{\dots }$	prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself.
${\sum }_p$, ${\prod }_p$	sum, product extended over all primes.
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${\sum }_{n\le x}$	${\sum }_{n=1}^{\lfloor x\rfloor }$.
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26: 26.18 Counting Techniques

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26.18.3 $$n!+\sum _{t=1}^n{(-1)}^t{r}_t(B)(n-t)!.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $r_j(B)$: number of rook positions and $B$: subset

Permalink:

http://dlmf.nist.gov/26.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

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27: 2.3 Integrals of a Real Variable

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2.3.7 $$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{a}_s{t}^{(s+\lambda -\mu )/\mu },$$ $t\to 0+$,

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Defines:

$q(t)$: function (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: positive constant, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(ii), §2.3(ii), §2.3(iv), §2.4(i), §2.4(ii)

Permalink:

http://dlmf.nist.gov/2.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

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2.3.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{x^{(s+\lambda )/\mu }},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Keywords:

Laplace transform

Referenced by:

§2.3(ii), §2.4(i)

Permalink:

http://dlmf.nist.gov/2.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

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2.3.12 $${\int }_0^{\mathrm{\infty }}f(xt)q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}ℳf\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{x^{(s+\lambda )/\mu }},$$ $x\to +\mathrm{\infty }$,

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\int$: integral, $f(x)$: function, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(ii), §2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

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As $t\to a+$

2.3.14

$$p(t)\sim p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: left endpoint, $p(t)$: real function, $p_s$: coefficients, $q(t)$: function, $q_s$: coefficients, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

item (b)

Permalink:

http://dlmf.nist.gov/2.3.E14

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_0>0$ (consistent with (a)).

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2.3.16 $$\frac{q(t)}{p^{\prime }(t)}\sim \sum _{s=0}^{\mathrm{\infty }}{b}_s{v}^{(s+\lambda -\mu )/\mu },$$ $v\to 0+$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $b$: right endpoint, $p(t)$: real function, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Permalink:

http://dlmf.nist.gov/2.3.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

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28: 26.12 Plane Partitions

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26.12.20 $$\sum _{\pi \subseteq \mathbb{N}\times \mathbb{N}\times \mathbb{N}}{q}^{|\pi |}=\prod _{k=1}^{\mathrm{\infty }}\frac{1}{{(1-q^k)}^k},$$

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Symbols:

$\mathbb{N}$: set of all positive integers, $\subseteq$: is, or is contained in, $k$: nonnegative integer, $\pi$: plane partition and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.21 $$\sum _{\pi \subseteq B(r,s,t)}{q}^{|\pi |}=\prod _{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+j+k-1}}{1-q^{h+j+k-2}}=\prod _{h=1}^r\prod _{j=1}^s\frac{1-q^{h+j+t-1}}{1-q^{h+j-1}},$$

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Symbols:

$\in$: element of, $\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $s$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.22

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Referenced by:

§26.12(ii)

Permalink:

http://dlmf.nist.gov/26.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.23

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.24

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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29: 27.12 Asymptotic Formulas: Primes

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27.12.4 $$\pi \left(x\right)\sim \sum _{k=1}^{\mathrm{\infty }}\frac{(k-1)!x}{{(\ln x)}^k}.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $k$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12 and Ch.27

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30: 25.16 Mathematical Applications

… ► $H\left(s\right)$ is analytic for $\mathrm{\Re }s>1$, and can be extended meromorphically into the half-plane $\mathrm{\Re }s>-2k$ for every positive integer $k$ by use of the relations …