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31: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►

§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

32: 16.19 Identities

§16.19 Identities

… ►This reference and Mathai (1993, §§2.2 and 2.4) also supply additional identities.

33: Mathematical Introduction

… ► ►►►

$ℂ$	complex plane (excluding infinity).
…
$\equiv$	equals by definition.
…

► ►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$𝐈$	unit matrix.
$mod$ or modulo	$m\equiv n(modp)$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$.
…

…

34: 17.12 Bailey Pairs

… ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

35: 20.4 Values at $z$ = 0

… ►

Jacobi’s Identity

…

36: 31.17 Physical Applications

… ►

$${𝐉}^2\mathrm{\Psi }(𝐱)\equiv {(𝐬+𝐭+𝐮)}^2\mathrm{\Psi }(𝐱)=j(j+1)\mathrm{\Psi }(𝐱),$$

►

$${𝐻}_s\mathrm{\Psi }(𝐱)\equiv (-2𝐬\cdot 𝐭-(2/a)𝐬\cdot 𝐮)\mathrm{\Psi }(𝐱)=h_s\mathrm{\Psi }(𝐱),$$

…

37: 1.2 Elementary Algebra

… ►The identity matrix $𝐈$, is defined as ►

1.2.53 $$𝐈=[{\delta }_{i,j}].$$

ⓘ

Defines:

$𝐈$: identity matrix

Symbols:

${\delta }_{j,k}$: Kronecker delta and $j$: integer

Permalink:

http://dlmf.nist.gov/1.2.E53

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

… ►

1.2.71 $$det(𝐀-\lambda 𝐈)=0,$$

ⓘ

Symbols:

$det$: determinant, $𝐈$: identity matrix and $𝐀$: non-defective matrix

Permalink:

http://dlmf.nist.gov/1.2.E71

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

… ►

1.2.72 $$(𝐀-\lambda 𝐈)𝐚=𝟎.$$

ⓘ

Symbols:

$𝐈$: identity matrix, $𝐀$: non-defective matrix and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§1.2(vi)

Permalink:

http://dlmf.nist.gov/1.2.E72

Encodings:

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See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

… ►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. …

38: 24.15 Related Sequences of Numbers

… ►

24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

►

24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

…

39: 25.4 Reflection Formulas

… ►

25.4.6 $$c\equiv -\ln \left(2\pi \right)-\frac{1}{2}\pi \mathrm{i}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $c$

Keywords:

definition

Source:

Apostol (1985a, p. 223)

Referenced by:

§25.6(ii)

Permalink:

http://dlmf.nist.gov/25.4.E6

Encodings:

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See also:

Annotations for §25.4 and Ch.25

40: 27.5 Inversion Formulas

… ►The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). …For example, the equation $\zeta \left(s\right)\cdot (1/\zeta \left(s\right))=1$ is equivalent to the identity …