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1: 1.1 Special Notation

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$x,y$	real variables.
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$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
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${𝐀}^{-1}$	inverse of the square matrix $𝐀$
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$det(𝐀)$	determinant of the square matrix $𝐀$
$\mathrm{tr}(𝐀)$	trace of the square matrix $𝐀$
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${𝐀}^{\ast }$	adjoint of the square matrix $𝐀$
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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.11

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E11

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Annotations for §1.18(ii), §1.18 and Ch.1

►Functions $f,g\in L^2(X,d\alpha )$ for which $⟨f-g,f-g⟩=0$ are identified with each other. The space $L^2(X,d\alpha )$ becomes a separable Hilbert space with inner product … ►

1.18.13 $$c_n=⟨f,{\varphi }_n⟩={\int }_a^{b}f(x)\overline{{\varphi }_n(x)}dx,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E13

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Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.52 $$⟨{\left(z-T\right)}^{-1}f,f⟩={\int }_{𝝈}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$ $f\in L^2\left(X\right)$, $z\notin {𝝈}_c$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral, $\notin$: not an element of, $z$: variable and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(vi), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E52

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Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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3: 1.2 Elementary Algebra

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§1.2(vi) Square Matrices

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Special Forms of Square Matrices

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Norms of Square Matrices

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Non-Defective Square Matrices

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4: 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). …

5: 26.12 Plane Partitions

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26.12.9 $${\left(\prod _{h=1}^r\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}\right)}^2;$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E9

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Annotations for §26.12(i), §26.12 and Ch.26

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26.12.10 $$\left(\prod _{h=1}^r\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}\right)\left(\prod _{h=1}^{r+1}\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}\right);$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E10

Encodings:

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See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.11 $$\left(\prod _{h=1}^{r+1}\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}\right)\left(\prod _{h=1}^r\prod _{j=1}^{s+1}\frac{h+j+t-1}{h+j-1}\right).$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E11

Encodings:

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See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.13 $$\prod _{h=1}^r\prod _{j=1}^r\frac{h+j+t-1}{h+j-1};$$

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Symbols:

$r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E13

Encodings:

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Annotations for §26.12(i), §26.12 and Ch.26

… ►where ${\sigma }_2(j)$ is the sum of the squares of the divisors of $j$. …

6: 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

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Annotations for §27.6 and Ch.27

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27.6.2 $$\sum _{d|n}\mu \left(d\right)f(d)=\prod _{p|n}(1-f(p)),$$ $n>1$.

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.6.E2

Encodings:

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Annotations for §27.6 and Ch.27

►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …

7: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$. …

8: 4.35 Identities

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§4.35(ii) Squares and Products

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9: 18.39 Applications in the Physical Sciences

… ►All are written in the same form as the product of three factors: the square root of a weight function $w(x)$, the corresponding OP or EOP, and constant factors ensuring unit normalization. …

10: 10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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