cash app phone number +1(888‒481‒4477)

SearchAdvancedHelp

(0.012 seconds)

21—30 of 61 matching pages

21: 13.8 Asymptotic Approximations for Large Parameters

… ►

13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differential equation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

…

22: 27.14 Unrestricted Partitions

… ►

27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

ⓘ

Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.6 $$p\left(n\right)=\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\left(p\left(n-\omega (k)\right)+p\left(n-\omega (-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+\mathrm{\cdots },$$

ⓘ

Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer and $\omega (k)$: pentagonal numbers

A&S Ref:

24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

…

23: 27.3 Multiplicative Properties

… ►

27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: 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

►

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

ⓘ

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

…

24: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

25: 24.2 Definitions and Generating Functions

… ►

$B_{2n+1}=0$ ,

►

${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

… ►

$E_{2n+1}=0$ ,

…

26: 4.13 Lambert $W$-Function

… ►

4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

ⓘ

Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

… ►

4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

ⓘ

Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

…

27: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7 and Ch.35

… ►

35.7.8 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b-c+\frac{1}{2}(m+1)};𝐈-𝐓),$$ ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(ii), Erratum (V1.0.26) for Equation (35.7.8)

Permalink:

http://dlmf.nist.gov/35.7.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.26):

Originally the constraint was written as $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$. The constraint has been replaced with ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$; for details see https://arxiv.org/abs/2002.05248.

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

…

28: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Let $p$ and $q$ be nonnegative integers; $a_1,\mathrm{\dots },a_p\in ℂ$; $b_1,\mathrm{\dots },b_q\in ℂ$; $-b_j+\frac{1}{2}(k+1)\notin \mathbb{N}$, $1\le j\le q$, $1\le k\le m$. … ►If $-a_j+\frac{1}{2}(k+1)\in \mathbb{N}$ for some $j,k$ satisfying $1\le j\le p$, $1\le k\le m$, then the series expansion (35.8.1) terminates. …

29: 24.14 Sums

… ►

24.14.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k{E}_{n-k}=-2^{n+1}{E}_{n+1}\left(0\right)=-2^{n+2}(1-2^{n+2})\frac{B_{n+2}}{n+2}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

… ►

24.14.8 $$\sum \frac{(2n)!}{(2j)!(2k)!(2\mathrm{\ell })!}{B}_{2j}{B}_{2k}{B}_{2\mathrm{\ell }}=(n-1)(2n-1)B_{2n}+n(n-\frac{1}{2})B_{2n-2},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $\mathrm{\ell }$: integer and $n$: integer

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(ii), §24.14 and Ch.24

… ►

24.14.11 $$det[B_{r+s}]={(-1)}^{n(n+1)/2}{\left(\prod _{k=1}^{n}k!\right)}^6/\left(\prod _{k=1}^{2n+1}k!\right),$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $det$: determinant, $!$: factorial (as in $n!$), $k$: integer, $n$: integer, $r$: row and $s$: column

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12 $$det[E_{r+s}]={(-1)}^{n(n+1)/2}{\left(\prod _{k=1}^{n}k!\right)}^2.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $det$: determinant, $!$: factorial (as in $n!$), $k$: integer, $n$: integer, $r$: row and $s$: column

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(ii), §24.14 and Ch.24

…

30: 25.6 Integer Arguments

… ►

25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

…