positive sums

SearchAdvancedHelp

(0.004 seconds)

11—20 of 141 matching pages

11: Bibliography

… ►

Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

ⓘ

External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.

Encodings:

BibTeX

…

12: 27.4 Euler Products and Dirichlet Series

… ►

27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

►

27.4.6 $$\sum _{n=1}^{\mathrm{\infty }}\varphi \left(n\right)n^{-s}=\frac{\zeta \left(s-1\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>2$,

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.2 I.B

Permalink:

http://dlmf.nist.gov/27.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

►

27.4.7 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)n^{-s}=\frac{\zeta \left(2s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\lambda \left(n\right)$: Liouville’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

… ►

27.4.10 $$\sum _{n=1}^{\mathrm{\infty }}{d}_k\left(n\right)n^{-s}={(\zeta \left(s\right))}^k,$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $d_k\left(n\right)$: divisor function, $\mathrm{\Re }$: real part, $k$: positive integer and $n$: positive integer

Referenced by:

§27.4

Permalink:

http://dlmf.nist.gov/27.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

►

27.4.11 $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)n^{-s}=\zeta \left(s\right)\zeta \left(s-\alpha \right),$$ $\mathrm{\Re }s>\max (1,1+\mathrm{\Re }\alpha )$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

…

13: Errata

… ►For some classical polynomials we give some positive sums and discriminants. …

14: 27.7 Lambert Series as Generating Functions

… ►

27.7.1 $$\sum _{n=1}^{\mathrm{\infty }}f(n)\frac{x^n}{1-x^n}.$$

ⓘ

Symbols:

$n$: positive integer and $x$: real number

Referenced by:

§27.7

Permalink:

http://dlmf.nist.gov/27.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

… ►

27.7.2 $$\sum _{n=1}^{\mathrm{\infty }}f(n)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}\sum _{d|n}f(d)x^n.$$

ⓘ

Symbols:

$d$: positive integer, $n$: positive integer and $x$: real number

Referenced by:

§27.7

Permalink:

http://dlmf.nist.gov/27.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

… ►

27.7.3 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)\frac{x^n}{1-x^n}=x,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

►

27.7.4 $$\sum _{n=1}^{\mathrm{\infty }}\varphi \left(n\right)\frac{x^n}{1-x^n}=\frac{x}{{(1-x)}^2},$$

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $n$: positive integer and $x$: real number

A&S Ref:

24.3.2 I.B

Permalink:

http://dlmf.nist.gov/27.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

►

27.7.5 $$\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha }\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)x^n,$$

ⓘ

Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

…

15: 26.9 Integer Partitions: Restricted Number and Part Size

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. …

16: 24.14 Sums

… ►In the next identity, valid for $n\ge 4$, the sum is taken over all positive integers $j,k,\mathrm{\ell },m$ with $j+k+\mathrm{\ell }+m=n$. …

17: 27.5 Inversion Formulas

… ►

27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

ⓘ

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

… ►

27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.3 $$g(n)=\sum _{d|n}f(d)⟺f(n)=\sum _{d|n}g(d)\mu \left(\frac{n}{d}\right).$$

ⓘ

Symbols:

$\mu \left(n\right)$: 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.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.4 $$n=\sum _{d|n}\varphi \left(d\right)⟺\varphi \left(n\right)=\sum _{d|n}d\mu \left(\frac{n}{d}\right),$$

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.2 II.B

Permalink:

http://dlmf.nist.gov/27.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

►

27.5.5 $$\ln n=\sum _{d|n}\mathrm{\Lambda }\left(d\right)⟺\mathrm{\Lambda }\left(n\right)=\sum _{d|n}(\ln d)\mu \left(\frac{n}{d}\right).$$

ⓘ

Symbols:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function, $\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.5.E5

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 §27.5 and Ch.27

…

18: 27.2 Functions

… ►

27.2.6 $${\varphi }_k\left(n\right)=\sum _{\left(m,n\right)=1}{m}^k,$$

ⓘ

Defines:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. ►

27.2.7 $$\varphi \left(n\right)={\varphi }_0\left(n\right).$$

ⓘ

Defines:

$\varphi \left(n\right)$: Euler’s totient

Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.10 $${\sigma }_{\alpha }\left(n\right)=\sum _{d|n}{d}^{\alpha },$$

ⓘ

Defines:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number

Symbols:

$d$: positive integer, $n$: positive integer and $\alpha$: parameter

A&S Ref:

24.3.3 I.A

Referenced by:

§27.14(ii)

Permalink:

http://dlmf.nist.gov/27.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

…

19: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.1 $$\sum _{n\le x}f(n)=F(x)+O\left(g(x)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

… ►

27.11.2 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11

Permalink:

http://dlmf.nist.gov/27.11.E2

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 §27.11 and Ch.27

… ►

27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E3

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 §27.11 and Ch.27

… ►

27.11.4 $$\sum _{n\le x}{\sigma }_1\left(n\right)=\frac{{\pi }^2}{12}{x}^2+O\left(x\ln x\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.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 §27.11 and Ch.27

►

27.11.5 $$\sum _{n\le x}{\sigma }_{\alpha }\left(n\right)=\frac{\zeta \left(\alpha +1\right)}{\alpha +1}{x}^{\alpha +1}+O\left(x^{\beta }\right),$$ $\alpha >0$, $\alpha \ne 1$, $\beta =\max (1,\alpha )$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

…

20: 26.15 Permutations: Matrix Notation

… ►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►

26.15.3 $$R(x,B)=\sum _{j=0}^n{r}_j(B)x^j.$$

ⓘ

Symbols:

$x$: real variable, $j$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

… ►

26.15.7 $$N(x,B)=\sum _{k=0}^n{r}_k(B)(n-k)!{(x-1)}^k.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

… ►

26.15.11 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=\prod _{j=1}^n(x+b_j-j+1).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

… ►

26.15.12 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=x^n,$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E12

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

…