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31: 1.10 Functions of a Complex Variable

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§1.10(ix) Infinite Products

… ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. … ►

Weierstrass Product

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§1.10(x) Infinite Partial Fractions

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32: 31.15 Stieltjes Polynomials

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31.15.4 $$G({\zeta }_1,{\zeta }_2,\mathrm{\dots },{\zeta }_n)=\prod _{k=1}^n\prod _{\mathrm{\ell }=1}^N{({\zeta }_k-a_{\mathrm{\ell }})}^{{\gamma }_{\mathrm{\ell }}/2}\prod _{j=k+1}^n({\zeta }_k-{\zeta }_j).$$

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Symbols:

$\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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§31.15(iii) Products of Stieltjes Polynomials

… ►The products …with respect to the inner product …The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho }^2(Q)$. …

33: 20.4 Values at $z$ = 0

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20.4.2 $${\theta }_1^{\prime }(0,q)=2q^{1/4}\prod _{n=1}^{\mathrm{\infty }}{\left(1-q^{2n}\right)}^3=2q^{1/4}{(q^2;q^2)}_{\mathrm{\infty }}^3,$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $n$: integer and $q$: nome

Referenced by:

Erratum (V1.0.28) for Equation (20.4.2)

Permalink:

http://dlmf.nist.gov/20.4.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.28):

The representation in terms of ${(q^2;q^2)}_{\mathrm{\infty }}^3$ was added to this equation.

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.3 $${\theta }_2(0,q)=2q^{1/4}\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n}\right)}^2,$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.3)

Permalink:

http://dlmf.nist.gov/20.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.4 $${\theta }_3(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n-1}\right)}^2,$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.2)

Permalink:

http://dlmf.nist.gov/20.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.5 $${\theta }_4(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1-q^{2n-1}\right)}^2.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

(20.10.1)

Permalink:

http://dlmf.nist.gov/20.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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Jacobi’s Identity

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34: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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35: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(ii) Integrals of Products with Bessel Functions

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§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

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28.28.25 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

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Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

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36: 16.17 Definition

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16.17.1 $$G_{p,q}^{m,n}(z;𝐚;𝐛)=G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{\mathrm{\ell }=1}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1-b_{\mathrm{\ell }+1}+s\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-s\right)\right)\right)z^{s}ds,$$

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Defines:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

Figure 16.17.1, Figure 16.17.1, §16.19

Permalink:

http://dlmf.nist.gov/16.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

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16.17.3 $$A_{p,q,k}^{m,n}(z)=\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne k\end{array}}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-b_k\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1+b_k-a_{\mathrm{\ell }}\right)z^{b_k}/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1+b_k-b_{\mathrm{\ell }+1}\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-b_k\right)\right).$$

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Defines:

$A_{p,q}^{m,n}(z)$: coefficient (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

37: Notices

… ►Certain products, commercial and otherwise, are mentioned in the DLMF. … ►NIST expressly does not endorse or recommend any specific product or service. …

38: 16.12 Products

§16.12 Products

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39: 26.13 Permutations: Cycle Notation

… ►Every permutation is a product of transpositions. A permutation with cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$ can be written as a product of $a_2+2a_3+\mathrm{\cdots }+(n-1)a_n=n-(a_1+a_2+\mathrm{\cdots }+a_n)$ transpositions, and no fewer. … ►Every transposition is the product of adjacent transpositions. If , then $\left(j,k\right)$ is a product of $2k-2j-1$ adjacent transpositions: …Every permutation is a product of adjacent transpositions. …

40: 25.2 Definition and Expansions

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§25.2(iv) Infinite Products

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25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infinite product, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

►product over all primes $p$. ►

25.2.12 $$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $s$: complex variable and $\rho$: zeros

Keywords:

infinite product, primes

Source:

Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31)

A&S Ref:

23.2.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

►product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).