quintuple product identity

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1: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

Jacobi’s Triple Product

… ►

Quintuple Product Identity

… ►Apart from Jacobi’s triple product identity (17.8.1) and the quintuple product identity (17.8.3) (see Cooper (2006) for a review), there also exist higher-order tuple product identities. One may see Pascadi (2021) for discussions and derivations of sextuple, septuple, octuple, nonuple and undecuple product identities. These identities are all given in terms of sums and products of basic bilateral hypergeometric series. …

2: Bibliography C

… ►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

ⓘ

External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

J. A. Cochran (1964) Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §10.21(x), §10.21(x).
Encodings:: BibTeX

… ►

J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:: MathReview (R. G. Langebartel), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

►

J. A. Cochran (1966b) The asymptotic nature of zeros of cross-product Bessel functions. Quart. J. Mech. Appl. Math. 19 (4), pp. 511–522.

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External Links:: ISSN 0033-5614, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

S. Cooper (2006) The quintuple product identity. Int. J. Number Theory 2 (1), pp. 115–161.

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External Links:: ISSN 1793-0421, Document, Link, MathReview (Andrew V. Sills)
Referenced by:: §17.8.
Encodings:: BibTeX

…

3: 24.10 Arithmetic Properties

… ►The denominator of

B_{2n}

is the product of all these primes

p

. … ►where

m\equiv n\not\equiv 0\pmod{p-1}

. …valid when

m\equiv n\pmod{(p-1)p^{\ell}}

and

n\not\equiv 0\pmod{p-1}

, where

\ell(\geq 0)

is a fixed integer. … ►valid for fixed integers

\ell(\geq 1)

, and for all

n(\geq 1)

such that

2n\not\equiv 0

\pmod{p-1}

and

p^{\ell}\mathbin{|}2n

. ►

24.10.9

E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(iv)
Permalink:: http://dlmf.nist.gov/24.10.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iv), §24.10 and Ch.24

…

4: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. … ►Their product

m

has 20 digits, twice the number of digits in the data. …

5: 1.1 Special Notation

… ► ►►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…
$\mathbf{I}$	identity matrix
…

…

6: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►The transpose of the product is … ►Column vectors

\mathbf{u}

and

\mathbf{v}

of the same length

n

have a scalar product … ►The scalar product has properties … ►The identity matrix

\mathbf{I}

, is defined as …

7: 27.16 Cryptography

… ►The primes are kept secret but their product

n=pq

, an 800-digit number, is made public. … ►Thus,

y\equiv x^{r}\pmod{n}

and

1\leq y<n

. … ►By the Euler–Fermat theorem (27.2.8),

x^{\phi\left(n\right)}\equiv 1\pmod{n}

; hence

x^{t\phi\left(n\right)}\equiv 1\pmod{n}

. But

y^{s}\equiv x^{rs}\equiv x^{1+t\phi\left(n\right)}\equiv x\pmod{n}

, so

y^{s}

is the same as

x

modulo

n

. …

8: 15.17 Mathematical Applications

… ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

9: 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. Every multiplicative

f

satisfies the identity …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 …

10: 24.19 Methods of Computation

… ►Another method is based on the identities ►

24.19.1

N_{2n}=\frac{2(2n)!}{(2\pi)^{2n}}\left(\prod_{p-1\mathbin{|}2n}p\right)\left(% \prod_{p}\frac{p^{2n}}{p^{2n}-1}\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : integer and $p$ : prime
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(i), §24.19 and Ch.24

►

D_{2n}=\prod_{p-1\mathbin{|}2n}p,

… ►If

\widetilde{N}_{2n}

denotes the right-hand side of (24.19.1) but with the second product taken only for

p\leq\left\lfloor(\pi e)^{-1}2n\right\rfloor+1

, then

N_{2n}=\left\lceil\widetilde{N}_{2n}\right\rceil

for

n\geq 2

. … ►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. …