quintuple product identity

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11: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates …called the Dirichlet product (or convolution) of $f$ and $g$. 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 … ►

27.5.8 $$g(n)=\prod _{d|n}f(d)⟺f(n)=\prod _{d|n}{\left(g\left(\frac{n}{d}\right)\right)}^{\mu \left(d\right)}.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E8

Encodings:

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See also:

Annotations for §27.5 and Ch.27

…

12: 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 $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. …

13: 20.4 Values at $z$ = 0

… ►

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

►

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:

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

►

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:

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

►

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:

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See also:

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

►

Jacobi’s Identity

…

14: 22.9 Cyclic Identities

§22.9 Cyclic Identities

… ►

§22.9(ii) Typical Identities of Rank 2

… ► ►

§22.9(iii) Typical Identities of Rank 3

… ►

15: 25.15 Dirichlet $L$-functions

… ► ►

25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

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Annotations for §25.15(i), §25.15 and Ch.25

►with the product taken over all primes $p$, beginning with $p=2$. … ►

25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

product representation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

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Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi \mathrm{i}r/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

definition

Source:

Apostol (1976, p. 262)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.6)

Permalink:

http://dlmf.nist.gov/25.15.E6

Encodings:

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Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(i), §25.15 and Ch.25

…

16: 27.14 Unrestricted Partitions

… ►Euler introduced the reciprocal of the infinite product … ►

§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity …implies $p\left(5n+4\right)\equiv 0(mod5)$. …

17: 25.10 Zeros

… ►The product representation (25.2.11) implies $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s>1$. … ►

25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

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See also:

Annotations for §25.10(i), §25.10 and Ch.25

… ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

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See also:

Annotations for §25.10(i), §25.10 and Ch.25

…

18: 27.8 Dirichlet Characters

… ►

27.8.6

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

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See also:

Annotations for §27.8 and Ch.27

►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. … ►

27.8.7 $$\chi \left(a\right)=1\text{ for all }a\equiv 1\text{ (mod }d\text{)},$$ $\left(a,k\right)=1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $\equiv$: modular equivalence, $d$: positive integer and $k$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E7

Encodings:

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See also:

Annotations for §27.8 and Ch.27

►Every Dirichlet character $\chi$ (mod $k$) is a product …

19: Bibliography R

… ►

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

… ►

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

►

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

… ►

Combinatorial Identities. Robert E. Krieger Publishing Co., Huntington, NY.

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Notes:

Reprint of the 1968 original with corrections.

External Links:

ISBN 0-88275-829-2, MathReview

Referenced by:

§24.5(iii), §26.5(iii), §26.8(i), §26.8(iv), §26.8(v).

Encodings:

BibTeX

… ►

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

…

20: 21.7 Riemann Surfaces

… ►

§21.7(ii) Fay’s Trisecant Identity

… ►where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). … ►

§21.7(iii) Frobenius’ Identity

…