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31: 3.3 Interpolation

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3.3.35 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\sum _{k=0}^n\left(f(z_k)/\prod _{\begin{array}{c}0\le j\le n\\ j\ne k\end{array}}(z_k-z_j)\right).$$

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Symbols:

$\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $f(z)$: function and $z_k$: nodes of function

A&S Ref:

25.1.5 (in slightly different form)

Permalink:

http://dlmf.nist.gov/3.3.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

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32: 18.30 Associated OP’s

… ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). …

33: 27.14 Unrestricted Partitions

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27.14.18 $$x\prod _{n=1}^{\mathrm{\infty }}{(1-x^n)}^{24}=\sum _{n=1}^{\mathrm{\infty }}\tau \left(n\right)x^n,$$ .

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Defines:

$\tau \left(n\right)$: Ramanujan’s tau function

Symbols:

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

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E18

Encodings:

pMML, png, TeX

See also:

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

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34: 5.14 Multidimensional Integrals

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5.14.2 $${\int }_{V_n}{\left(1-\sum _{k=1}^n{t}_k\right)}^{z_{n+1}-1}\prod _{k=1}^n{t}_k^{z_k-1}d{t}_k=\frac{\mathrm{\Gamma }\left(z_1\right)\mathrm{\Gamma }\left(z_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(z_{n+1}\right)}{\mathrm{\Gamma }\left(z_1+z_2+\mathrm{\cdots }+z_{n+1}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $k$: nonnegative integer, $z$: complex variable and $V_n$: simplex

Referenced by:

§5.19(iii)

Permalink:

http://dlmf.nist.gov/5.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14 and Ch.5

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35: 14.18 Sums

§14.18 Sums

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§14.18(iii) Other Sums

… ►For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22). … ►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). …

36: 25.8 Sums

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25.8.4 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (2.7), p. 135)

Permalink:

http://dlmf.nist.gov/25.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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37: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …It can be expressed as a sum over all primes $p\le x$: … ►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …

38: 20.5 Infinite Products and Related Results

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20.5.9 $${\theta }_3\left(\pi z\right|\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{p}^{2n}{q}^{n^2}=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+q^{2n-1}{p}^2\right)\left(1+q^{2n-1}{p}^{-2}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

Referenced by:

§17.8, §20.12(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5(i), §20.5 and Ch.20

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39: 20.12 Mathematical Applications

… ►For applications of ${\theta }_3(0,q)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …

40: 19.23 Integral Representations

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19.23.9 $$R_{-a}(𝐛;𝐳)=\frac{4\mathrm{\Gamma }\left(b_1+b_2+b_3\right)}{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right)}{\int }_0^{\pi /2}{\int }_0^{\pi /2}{\left(\sum _{j=1}^3{z}_j{l}_j^2\right)}^{-a}\prod _{j=1}^3{l}_j^{2b_j-1}\sin \theta d\theta d\varphi ,$$ $b_j>0$, $\mathrm{\Re }{z}_j>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $l$: nonnegative integer and $\varphi$: real or complex argument

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.10 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^1{u}^{a-1}{(1-u)}^{a^{\prime }-1}\prod _{j=1}^n{(1-u+u{z}_j)}^{-b_j}du,$$ $a,a^{\prime }>0$; $a+a^{\prime }={\sum }_{j=1}^n{b}_j$; $z_j\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.19, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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