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1: 30.10 Series and Integrals

… ►For product formulas and convolutions see Connett et al. (1993). …For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

2: 13.12 Products

§13.12 Products

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).

3: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►On ${ℝ}^d$ consider the weight function $\exp \left(-{\Vert 𝐱\Vert }^2\right)$ and the corresponding inner product …The OPs of degree $n$ with respect to the inner product (37.17.1) form the space ${𝒱}_n\left({ℝ}^d\right)$. … ►

§37.17(i) Product Hermite Polynomials

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§37.17(vi) Hermite Polynomials for Weight Function ${\mathrm{e}}^{-⟨𝐀𝐱,𝐱⟩}$

►For a positive definite symmetric $d\times d$ matrix $𝐀$ define the inner product …

4: 13.25 Products

§13.25 Products

… ►For integral representations, integrals, and series containing products of $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).

5: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … ►Euler products are used to find series that generate many functions of multiplicative number theory. …

6: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

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4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

…

7: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

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4.36.1 $$\sinh z=z\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.2 $$\cosh z=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

…

8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. With norm defined by …Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►thus generalizing the inner product of (1.18.9). … ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. …

9: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

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§20.5(i) Single Products

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Jacobi’s Triple Product

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§20.5(iii) Double Products

… ►

10: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►where $v_{\mathrm{\ell }}$ is the $\mathrm{\ell }$th component of $𝐯$ and $𝐱{\sigma }_{𝐯}$ denotes the reflection $𝐱{\sigma }_{𝐯}=𝐱-2\frac{⟨𝐱,𝐯⟩}{⟨𝐯,𝐯⟩}𝐯.$ These operators commute; that is, $T_{\mathrm{\ell }}{T}_j=T_j{T}_{\mathrm{\ell }}$ for . …They are orthogonal with respect to the inner product … ► … ►These are orthogonal polynomials for an inner product that involves derivatives of functions; see Marcellán and Xu (2015) for the one-variable case. There are many such inner products. …