positive sums

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1: 18.14 Inequalities

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§18.14(iv) Positive Sums

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Jacobi

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2: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

3: Bibliography G

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Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

(18.14.25), (18.14.26), (18.14.27).

Encodings:

BibTeX

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4: 27.10 Periodic Number-Theoretic Functions

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27.10.4 $$c_k\left(n\right)=\sum _{m=1}^k{\chi }_1\left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Defines:

$c_k\left(n\right)$: Ramanujan’s sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.5 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.9 $$G(n,\chi )=\sum _{m=1}^k\chi \left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k}.$$

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Defines:

$G(n,\chi )$: Gauss sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.10 $$G(n,\chi )=\overline{\chi }(n)G(1,\chi ).$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\overline{z}$: complex conjugate, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.11 $${|G(1,\chi )|}^2=k.$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $k$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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5: 26.2 Basic Definitions

… ►A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. …

6: 26.10 Integer Partitions: Other Restrictions

… ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

7: 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 … ►

27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

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

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27.14.10 $$A_k(n)=\sum _{\begin{array}{c}h=1\\ \left(h,k\right)=1\end{array}}^k\exp \left(\pi \mathrm{i}s(h,k)-2\pi \mathrm{i}n\frac{h}{k}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $k$: positive integer, $n$: positive integer, $A_k(n)$: coefficients and $s(h,k)$: Dedekind sum

A&S Ref:

24.2.1 I.C

Permalink:

http://dlmf.nist.gov/27.14.E10

Encodings:

pMML, png, TeX

See also:

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

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

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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

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27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

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Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

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

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8: 27.6 Divisor Sums

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27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.3 $$\sum _{d|n}|\mu \left(d\right)|=2^{\nu \left(n\right)},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\nu \left(n\right)$: number of distinct primes dividing a number, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.6 $$\sum _{d|n}{\varphi }_k\left(d\right){\left(\frac{n}{d}\right)}^k=1^k+2^k+\mathrm{\cdots }+n^k,$$

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Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.7 $$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^k=J_k\left(n\right),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $\mu \left(n\right)$: Möbius function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.8 $$\sum _{d|n}{J}_k\left(d\right)=n^k.$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

9: 26.14 Permutations: Order Notation

… ►Equivalently, this is the sum over of the number of integers less than $\sigma (j)$ that lie in positions to the right of the $j$th position: $inv(35247816)=2+3+1+1+2+2+0=11.$ … ►The major index is the sum of all positions that mark the first element of a descent: …

10: 27.13 Functions

… ►The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. … ►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. … ►

27.13.4 $$\vartheta \left(x\right)=1+2\sum _{m=1}^{\mathrm{\infty }}{x}^{m^2},$$ .

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Defines:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation

Symbols:

${\theta }_j(z,q)$: theta function, $m$: positive integer and $x$: real number

A&S Ref:

16.27.3 (with $z=0$, $q=x$)

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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27.13.6 $${(\vartheta \left(x\right))}^2=1+4\sum _{n=1}^{\mathrm{\infty }}\left({\delta }_1(n)-{\delta }_3(n)\right)x^n,$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $n$: positive integer, $x$: real number and ${\delta }_j(n)$: number of divisors

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

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