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31: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►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 $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

32: 14.28 Sums

§14.28 Sums

… ►

14.28.1 $$P_{\nu }\left(z_1{z}_2-{\left(z_1^2-1\right)}^{1/2}{\left(z_2^2-1\right)}^{1/2}\cos \varphi \right)=P_{\nu }\left(z_1\right)P_{\nu }\left(z_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(z_1\right)P_{\nu }^m(z_2)\cos \left(m\varphi \right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cos z$: cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $z$: complex variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(i), §14.28 and Ch.14

… ►

14.28.2 $$\sum _{n=0}^{\mathrm{\infty }}(2n+1)Q_n\left(z_1\right)P_n\left(z_2\right)=\frac{1}{z_1-z_2},$$ $z_1\in {ℰ}_1$, $z_2\in {ℰ}_2$,

ⓘ

Symbols:

$\in$: element of, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $z$: complex variable, $n$: nonnegative integer and ${ℰ}_1$, ${ℰ}_2$: ellipses

Permalink:

http://dlmf.nist.gov/14.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(ii), §14.28 and Ch.14

… ►

§14.28(iii) Other Sums

…

33: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►and $s(h,k)$ is a Dedekind sum given by ►

27.14.11 $$s(h,k)=\sum _{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\lfloor \frac{hr}{k}\rfloor -\frac{1}{2}\right).$$

ⓘ

Defines:

$s(h,k)$: Dedekind sum (locally)

Symbols:

$\lfloor x\rfloor$: floor of $x$ and $k$: positive integer

A&S Ref:

24.2.1 I.C

Referenced by:

§23.18, §27.14(iv)

Permalink:

http://dlmf.nist.gov/27.14.E11

Encodings:

pMML, png, TeX

See also:

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

… ►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by …where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). …

34: 31.14 General Fuchsian Equation

… ►

31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

… ►

$\alpha +\beta +1={\displaystyle \sum _{j=1}^N}{\gamma }_j,$

►

$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

… ►

31.14.4 $$\frac{d^{2}W}{{dz}^2}=\sum _{j=1}^N\left(\frac{{\tilde{\gamma }}_j}{{(z-a_j)}^2}+\frac{{\tilde{q}}_j}{z-a_j}\right)W,$$ ${\sum }_{j=1}^N{\tilde{q}}_j=0$,

ⓘ

Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

►

${\tilde{q}}_j={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^N}{\displaystyle \frac{{\gamma }_j{\gamma }_k}{a_j-a_k}}-q_j,$

…

35: 6.6 Power Series

… ►

6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

… ►

6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

ⓘ

Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

…

36: 10.53 Power Series

… ►

10.53.1 $${𝗃}_n\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.2

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.2 $${𝗒}_n\left(z\right)=-\frac{1}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(\frac{1}{2}{z}^2)}^k}{k!}+\frac{{(-1)}^{n+1}}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.3

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

…

37: 26.18 Counting Techniques

… ►

26.18.1

ⓘ

Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

… ►

26.18.2

ⓘ

Symbols:

$\lfloor x\rfloor$: floor of $x$ , $j$: nonnegative integer, $n$: nonnegative integer, $N$: set and $p_n$: primes

Permalink:

http://dlmf.nist.gov/26.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

… ►

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

ⓘ

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

… ►

26.18.4 $$k^n+\sum _{t=1}^n{(-1)}^t\left(\genfrac{}{}{0.0pt}{}{k}{t}\right){(k-t)}^n.$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

…

38: 20.6 Power Series

… ►

20.6.6 $${\delta }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{\begin{array}{c}m=-\mathrm{\infty }\\ \left|m\right|+\left|n\right|\ne 0\end{array}}^{\mathrm{\infty }}{(m+n\tau )}^{-2j},$$

ⓘ

Defines:

${\delta }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Referenced by:

§20.6, §20.6

Permalink:

http://dlmf.nist.gov/20.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.7 $${\alpha }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+n\tau )}^{-2j},$$

ⓘ

Defines:

${\alpha }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.8 $${\beta }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )}^{-2j},$$

ⓘ

Defines:

${\beta }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.9 $${\gamma }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m+(n-\frac{1}{2})\tau )}^{-2j},$$

ⓘ

Defines:

${\gamma }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

… ►For further information on ${\delta }_{2j}$ see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have ${\delta }_{2n}=c_n/(2n-1)$ when $n\ge 2$.

39: 2.10 Sums and Sequences

§2.10 Sums and Sequences

►

§2.10(i) Euler–Maclaurin Formula

… ► … ►

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►See also Flajolet and Odlyzko (1990).

40: 14.18 Sums

§14.18 Sums

… ►

§14.18(iii) Other Sums

… ►

14.18.6 $$(x-y)\sum _{k=0}^n(2k+1)P_k\left(x\right)P_k\left(y\right)=(n+1)\left(P_{n+1}\left(x\right)P_n\left(y\right)-P_n\left(x\right)P_{n+1}\left(y\right)\right),$$

ⓘ

Symbols:

$P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $n$: nonnegative integer

A&S Ref:

8.9.1

Referenced by:

Erratum (V1.0.7) for Subsection 14.18(iii)

Permalink:

http://dlmf.nist.gov/14.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

… ►For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …