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11: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The minor $M_{jk}$ of the entry $a_{jk}$ in the $n$th-order determinant $det[a_{jk}]$ is the ($n-1$)th-order determinant derived from $det[a_{jk}]$ by deleting the $j$th row and the $k$th column. … ►

1.3.9 $$det{[a_{jk}]}^2\le \left(\sum _{k=1}^n{a}_{1k}^2\right)\left(\sum _{k=1}^n{a}_{2k}^2\right)\mathrm{\dots }\left(\sum _{k=1}^n{a}_{nk}^2\right).$$

ⓘ

Symbols:

$det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►

1.3.10 $$a_{j1}{a}_{k1}+a_{j2}{a}_{k2}+\mathrm{\cdots }+a_{jn}{a}_{kn}=0$$

ⓘ

Symbols:

$j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant …

12: 5.16 Sums

… ►

5.16.1 $$\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{\psi }^{\prime }\left(k\right)=-\frac{{\pi }^2}{8},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

►

5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

…

13: 25.2 Definition and Expansions

… ►

25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

ⓘ

Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

… ►

25.2.7 $${\zeta }^{(k)}\left(s\right)={(-1)}^k\sum _{n=2}^{\mathrm{\infty }}{(\ln n)}^k{n}^{-s},$$ $\mathrm{\Re }s>1$, $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Apostol (1976, p. 236); with $f(n)=1$

Permalink:

http://dlmf.nist.gov/25.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

… ►

25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

…

14: 29.5 Special Cases and Limiting Forms

… ►

29.5.4 $$\underset{k\to 1-}{lim}{a}_{\nu }^m\left(k^2\right)=\underset{k\to 1-}{lim}{b}_{\nu }^{m+1}\left(k^2\right)=\nu (\nu +1)-{\mu }^2,$$

ⓘ

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

►

29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

►

29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

… ►If $k\to 0+$ and $\nu \to \mathrm{\infty }$ in such a way that $k^2\nu (\nu +1)=4\theta$ (a positive constant), then ►

$lim{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$

…

15: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). … ►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …

16: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►

5.17.3 $$G\left(z+1\right)={(2\pi )}^{z/2}\exp \left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma z^2\right)\prod _{k=1}^{\mathrm{\infty }}\left({\left(1+\frac{z}{k}\right)}^k\exp \left(-z+\frac{z^2}{2k}\right)\right).$$

ⓘ

Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

… ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

ⓘ

Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

… ►

5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

…

17: 5.19 Mathematical Applications

… ►

$a_k={\displaystyle \frac{k}{(3k+2)(2k+1)(k+1)}}.$

… ►

5.19.2 $$a_k=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(\frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+\frac{2}{3}}\right).$$

ⓘ

Symbols:

$k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

…

18: 19.3 Graphics

… ►

►

►

Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. …If ${\sin }^2\varphi =1/k^2$ (), then it has the value $K\left(1/k\right)/k$: put $c=k^2$ in (19.25.5) and use (19.25.1). Magnify 3D Help

►

►

►

Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. …If ${\sin }^2\varphi =1/k^2$ (), then it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$: put $c=k^2$ in (19.25.7) and use (19.25.1). Magnify 3D Help

… ►

►

►

Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …If ${\sin }^2\varphi =1/k^2$ (), then by (19.7.4) it reduces to $\mathrm{\Pi }(2/k^2,1/k)/k$, $k^2\ne 2$, with Cauchy principal value $(K\left(1/k\right)-\mathrm{\Pi }(\frac{1}{2},1/k))/k$, , by (19.6.5). … Magnify 3D Help

… ►

►

►

Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2\ge 1$) it is infinite at $k^2=1$, and has the value $K(1/k)/k$ when $k^2>1$. Magnify 3D Help

… ►

►

►

Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2>1$) it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$. Magnify 3D Help

…

19: 29.7 Asymptotic Expansions

… ►

29.7.3 $${\tau }_0=\frac{1}{2^3}(1+k^2)(1+p^2),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►The same Poincaré expansion holds for $b_{\nu }^{m+1}\left(k^2\right)$, since … ►

29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.8 $${\tau }_4=\frac{1}{2^{16}}({(1+k^2)}^5(63p^6+1260p^4+2943p^2+486)-8k^2{(1+k^2)}^3(49p^6+1010p^4+2493p^2+432)+16k^4(1+k^2)(35p^6+760p^4+2043p^2+378)).$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

…

20: 26.9 Integer Partitions: Restricted Number and Part Size

… ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. … ►

26.9.5 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k\left(n\right)q^n=\prod _{j=1}^k\frac{1}{1-q^j}=1+\sum _{m=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{k+m-1}{m}\right]}_q{q}^m,$$

ⓘ

Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.7 $$\sum _{m,n=0}^{\mathrm{\infty }}{p}_k(\le m,n)x^k{q}^n=1+\sum _{k=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q{x}^k=\prod _{j=0}^m\frac{1}{1-x{q}^j}.$$

ⓘ

Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.8 $$p_k\left(n\right)=p_k\left(n-k\right)+p_{k-1}\left(n\right);$$

ⓘ

Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.10(iii)

Permalink:

http://dlmf.nist.gov/26.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iii), §26.9 and Ch.26

… ►

26.9.10 $$p_k\left(n\right)\sim \frac{n^{k-1}}{k!(k-1)!}.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $!$: factorial (as in $n!$), $p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iv), §26.9 and Ch.26