representation as sums of powers

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1: 24.4 Basic Properties

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§24.4(iii) Sums of Powers

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2: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►

$S={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k,$

… ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

3: 11.9 Lommel Functions

… ►where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by ►

11.9.3 $$s_{\mu ,\nu }\left(z\right)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )},$$

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Defines:

$s_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Permalink:

http://dlmf.nist.gov/11.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.4 $$a_k(\mu ,\nu )=\prod _{m=1}^k\left({(\mu +2m-1)}^2-{\nu }^2\right)=4^k{\left(\frac{\mu -\nu +1}{2}\right)}_k{\left(\frac{\mu +\nu +1}{2}\right)}_k,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/11.9.E4

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

An alternative Pochhammer symbol representation was added.

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).

4: 27.13 Functions

… ►Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. … ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. … ►Hardy and Littlewood (1925) conjectures that when $k$ is not a power of 2, and that $G\left(k\right)\le 4k$ when $k$ is a power of 2, but the most that is known (in 2009) is for some constant $c$. … ►

§27.13(iv) Representation by Squares

… ►Mordell (1917) notes that $r_k\left(n\right)$ is the coefficient of $x^n$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$. …

5: 16.11 Asymptotic Expansions

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$c_k=-{\displaystyle \frac{1}{k{\kappa }^{\kappa }}}{\displaystyle \sum _{m=0}^{k-1}}{c}_m{e}_{k,m},$ $k\ge 1$,

… ►Explicit representations for the coefficients $c_k$ are given in Volkmer (2023). ►It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_j,-a_j-1,\mathrm{\dots }$, $j=1,\mathrm{\dots },p$, provided that these poles are all simple, that is, no two of the $a_j$ differ by an integer. … ►Here the upper or lower signs are chosen according as $z$ lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of $z{e}^{\mp \pi i}$ its phases are $\mathrm{ph}z\mp \pi$, respectively. … ►Explicit representations for the coefficients $c_k$ are given in Volkmer and Wood (2014). …

6: 7.17 Inverse Error Functions

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§7.17(ii) Power Series

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7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

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Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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7.17.2_5 $$a_{m+1}=\frac{1}{2m+3}\sum _{n=0}^m\frac{2n+1}{m-n+1}{a}_n{a}_{m-n},$$ $m=0,1,2,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer

Referenced by:

§7.17(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

… ►For an alternative representation of (7.17.3) see Blair et al. (1976).

7: 9.12 Scorer Functions

… ►where … ►

9.12.17 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1).

Referenced by:

(9.12.15), (9.12.18)

Permalink:

http://dlmf.nist.gov/9.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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§9.12(vii) Integral Representations

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9.12.29 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k}+\frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$.

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Permalink:

http://dlmf.nist.gov/9.12.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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8: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

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Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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9: 27.4 Euler Products and Dirichlet Series

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27.4.1 $$\sum _{n=1}^{\mathrm{\infty }}f(n)=\prod _p\left(1+\sum _{r=1}^{\mathrm{\infty }}f(p^r)\right),$$

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Symbols:

$n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $f$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.2 $$\sum _{n=1}^{\mathrm{\infty }}f(n)=\prod _p{(1-f(p))}^{-1}.$$

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Symbols:

$n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $f$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

… ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): … ►

27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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

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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

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10: 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 … ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula … ►and $s(h,k)$ is a Dedekind sum given 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). … ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: …