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11: Bibliography C

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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.

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External Links:

MathReview (T. Hayashida)

Referenced by:

§27.13(ii).

Encodings:

BibTeX

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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

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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

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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

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Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

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External Links:

ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt

Referenced by:

§30.10, §30.11(vi).

Encodings:

BibTeX

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12: 20 Theta Functions

Chapter 20 Theta Functions

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13: 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

Encodings:

pMML, png, TeX

See also:

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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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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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26.12.14 $$\prod _{h=1}^r\prod _{j=1}^{r+1}\frac{h+j+t-1}{h+j-1}.$$

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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.E14

Encodings:

pMML, png, TeX

See also:

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

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14: Foreword

… ►The production of these new resources has been a very complex undertaking some 10 years in the making. … ►November 20, 2009 …

15: 7.21 Physical Applications

… ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …

16: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, ►

27.2.1 $$n=\prod _{r=1}^{\nu \left(n\right)}{p}_r^{a_r},$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Referenced by:

§27.2(i), §27.3

Permalink:

http://dlmf.nist.gov/27.2.E1

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. … ►

Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
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17: 1.14 Integral Transforms

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Convolution

►For Fourier transforms, the convolution $(f\ast g)(t)$ of two functions $f(t)$ and $g(t)$ defined on $(-\mathrm{\infty },\mathrm{\infty })$ is given by … ►

Convolution

►For Laplace transforms, the convolution of two functions $f(t)$ and $g(t)$, defined on $[0,\mathrm{\infty })$, is … ►

Convolution

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18: 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. …

19: 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:

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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:

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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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20: Publications

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Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations in the NIST Digital Library of Mathematical Functions, in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds), Springer, Berlin, Heidelberg (2010) pp. 385–393.

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NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.