representations as sums of powers

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11: 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 $\tau \left(n\right)$: …

12: 18.3 Definitions

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots },6$ are given in §18.5(iv). For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►When $j=k=0$ the sum in (18.3.1) is $N+1$. …

13: 12.14 The Function $W(a,x)$

… ►

§12.14(v) Power-Series Expansions

… ►

12.14.9 $$w_1(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(a)\frac{x^{2n}}{(2n)!},$$

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Defines:

$w_1(a,x)$: even solution (locally)

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\alpha }_n(a)$: recursion

A&S Ref:

19.16.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E9

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Annotations for §12.14(v), §12.14 and Ch.12

►

12.14.10 $$w_2(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n(a)\frac{x^{2n+1}}{(2n+1)!},$$

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Defines:

$w_2(a,x)$: odd solution (locally)

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\beta }_n(a)$: recursion

A&S Ref:

19.16.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E10

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Annotations for §12.14(v), §12.14 and Ch.12

… ►

§12.14(vi) Integral Representations

… ►

12.14.29 $$l(\mu )\sim \frac{2^{\frac{1}{4}}}{{\mu }^{\frac{1}{2}}}\sum _{s=0}^{\mathrm{\infty }}\frac{l_s}{{\mu }^{4s}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $l(\mu )$ and $l_s$: coefficient

Permalink:

http://dlmf.nist.gov/12.14.E29

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Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

…

14: 8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(iii) Integral Representations

… ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. … ►

Power-Series Expansions

… ►

8.21.21 $$\mathrm{ci}(a,z)=-f(a,z)\sin z+g(a,z)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E21

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Annotations for §8.21(vii), §8.21 and Ch.8

… ►

8.21.26 $$f(a,z)\sim z^{a-1}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(1-a\right)}_{2k}}{z^{2k}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $f(a,z)$: auxiliary function

Permalink:

http://dlmf.nist.gov/8.21.E26

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Annotations for §8.21(viii), §8.21 and Ch.8

…

15: 13.29 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. … ►

§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. … ►

13.29.3 $${\mathrm{e}}^{-\frac{1}{2}z}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2\mu \right)}_s{\left(\frac{1}{2}+\mu -\kappa \right)}_s}{{\left(2\mu \right)}_{2s}s!}{(-z)}^{s}y(s),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable and $y(n)$: solution

Permalink:

http://dlmf.nist.gov/13.29.E3

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Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►

13.29.7 $$z^{-a}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a-b+1\right)}_s}{s!}w(s),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E7

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Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

…

16: 5.9 Integral Representations

§5.9 Integral Representations

… ►(The fractional powers have their principal values.) … ►For additional representations see Whittaker and Watson (1927, §§12.31–12.32). … ► …

17: Bibliography

… ►

Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.

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External Links:

ISSN 0044-2275, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.61(v), §10.64.

Encodings:

BibTeX

… ►

Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

… ►

Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

BibTeX

… ►

Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

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External Links:

Document, MathReview (D. L. Colton), ZentralBlatt

Referenced by:

§18.17(iv).

Encodings:

BibTeX

… ►

Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:

ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

…

18: 2.4 Contour Integrals

… ►

2.4.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }}$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\lambda$: constant, $a$: left endpoint, $q(t)$: analytic function and $\mu$: positive constant

Referenced by:

§2.4(i)

Permalink:

http://dlmf.nist.gov/2.4.E1

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Annotations for §2.4(i), §2.4 and Ch.2

… ►

2.4.4 $$Q(z)\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }},$$ $z\to \mathrm{\infty }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\lambda$: constant, $a$: left endpoint, $\mu$: positive constant and $Q(z)$: Laplace transform of $q(t)$

Referenced by:

§2.4(ii)

Permalink:

http://dlmf.nist.gov/2.4.E4

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Annotations for §2.4(ii), §2.4 and Ch.2

… ►

In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\lambda$: constant, $a$: left endpoint, $p(t)$: analytic function, $q(t)$: analytic function, $p_s$: coefficients, $q_s$: coefficients and $\mu$: positive constant

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E11

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Annotations for §2.4(iii), §2.4 and Ch.2

with $\mathrm{\Re }\lambda >0$, $\mu >0$, $p_0\ne 0$, and the branches of ${(t-a)}^{\lambda }$ and ${(t-a)}^{\mu }$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega$ as $t\to a$ along $𝒫$.

… ►However, if $p^{\prime }(t_0)=0$, then $\mu \ge 2$ and different branches of some of the fractional powers of $p_0$ are used for the coefficients $b_s$; again see §2.3(iii). … ►For integral representations of the $b_{2s}$ and their asymptotic behavior as $s\to \mathrm{\infty }$ see Boyd (1995). …

19: Bibliography G

… ►

Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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External Links:

ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.18.

Encodings:

BibTeX

… ►

Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

… ►

A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

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External Links:

ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

… ►

Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

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External Links:

Document, MathReview (E. A. Thieleker), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

… ►

Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

…

20: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►

19.36.13 $$2R_G(t_0^2,t_0^2+\theta c_0^2,t_0^2+\theta a_0^2)=\left(t_0^2+\theta \sum _{m=0}^{\mathrm{\infty }}{2}^{m-1}{c}_m^2\right)R_C(T^2+\theta M^2,T^2)+h_0+\sum _{m=1}^{\mathrm{\infty }}{2}^m(h_m-h_{m-1}).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $m$: nonnegative integer, $M$: limit, $T$: limit and $h_n$

Permalink:

http://dlmf.nist.gov/19.36.E13

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Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

… ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … ►Faster convergence of power series for $K\left(k\right)$ and $E\left(k\right)$ can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …