DLMF: Untitled Document

… ►

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.

ⓘ

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.

ⓘ

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.

ⓘ

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

… ►

W. C. Connett, C. Markett, and A. L. Schwartz (1993) Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt
Referenced by:: §30.10, §30.11(vi).
Encodings:: BibTeX

…

… ►where the last right-hand side is the sum over

m\geq 0

of the generating functions for partitions into distinct parts with largest part equal to

m

. … ►where the inner sum is the sum of all positive odd divisors of

t

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

k

for which

n-(3k^{2}\pm k)\geq 0

. … ►where the inner sum is the sum of all positive divisors of

t

that are in

S

. …

… ►

26.14.1

\mathop{\mathrm{inv}}(\sigma)=\sum_{\begin{subarray}{c}1\leq j<k\leq n\\[1.5pt% ] \sigma(j)>\sigma(k)\end{subarray}}1.

ⓘ

Symbols:: $j$ : nonnegative integer, $k$ : nonnegative integer, $n$ : nonnegative integer and $\sigma$ : permutation
Permalink:: http://dlmf.nist.gov/26.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(i), §26.14 and Ch.26

►Equivalently, this is the sum over

1\leq j<n

of the number of integers less than

\sigma(j)

that lie in positions to the right of the

j

th position:

\mathop{\mathrm{inv}}(35247816)=2+3+1+1+2+2+0=11.

… ►The major index is the sum of all positions that mark the first element of a descent: ►

26.14.2

\mathop{\mathrm{maj}}(\sigma)=\sum_{\begin{subarray}{c}1\leq j<n\\ \sigma(j)>\sigma(j+1)\end{subarray}}j.

ⓘ

Symbols:: $j$ : nonnegative integer, $n$ : nonnegative integer, $\sigma$ : permutation and $\mathop{\mathrm{maj}}(\sigma)$ : major index
Permalink:: http://dlmf.nist.gov/26.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(i), §26.14 and Ch.26

… ►

26.14.3

\sum_{\sigma\in\mathfrak{S}_{n}}q^{\mathop{\mathrm{inv}}(\sigma)}=\sum_{\sigma% \in\mathfrak{S}_{n}}q^{\mathop{\mathrm{maj}}(\sigma)}=\prod_{j=1}^{n}\frac{1-q% ^{j}}{1-q}.

ⓘ

Symbols:: $\in$ : element of, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $j$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(ii), §26.14 and Ch.26

…

… ►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, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes

p\leq x

: … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►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. …

… ►and the coefficients

\mathsf{A}_{s}(\tau)

are the product of

\tau^{s}

and a polynomial in

\tau

of degree

2s

. … ►

12.10.33

\mathsf{A}_{s+1}(\tau)=-4\tau^{2}(\tau+1)^{2}\frac{\mathrm{d}}{\mathrm{d}\tau}% \mathsf{A}_{s}(\tau)-\frac{1}{4}\int_{0}^{\tau}\left(20u^{2}+20u+3\right)% \mathsf{A}_{s}(u)\,\mathrm{d}u,

s=0,1,2,\dots

,

ⓘ

Defines:: $\mathsf{A}_{s}(\tau)$ : coefficients (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $s$ : nonnegative integer and $\tau$
Permalink:: http://dlmf.nist.gov/12.10.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vi), §12.10 and Ch.12

… ►

\mathsf{A}_{1}(\tau)=-\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9),

… ►

A_{s}(\zeta)=\zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m% }(t),

►

\zeta^{2}B_{s}(\zeta)=-\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(% 2s-m+1)}u_{2s-m+1}(t),

…

… ►The notation

\sum_{\pi\subseteq B(r,s,t)}

denotes the sum over all plane partitions contained in

B(r,s,t)

, and

|\pi|

denotes the number of elements in

\pi

. … ►

26.12.21

\sum_{\pi\subseteq B(r,s,t)}q^{|\pi|}=\prod_{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+% j+k-1}}{1-q^{h+j+k-2}}=\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{1-q^{h+j+t-1}}{1-q^% {h+j-1}},

ⓘ

Symbols:: $\in$ : element of, $\subseteq$ : is, or is contained in, $h$ : nonnegative integer, $j$ : nonnegative integer, $k$ : nonnegative integer, $\pi$ : plane partition, $r$ : positive integer, $s$ : positive integer, $t$ : positive integer, $B(r,s,t)$ : Box and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.12.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.22

\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,t)\\ \pi\mbox{\scriptsize\ symmetric}\end{subarray}}q^{|\pi|}=\prod_{h=1}^{r}\frac{% 1-q^{2h+t-1}}{1-q^{2h-1}}\prod_{1\leq h<j\leq r}\frac{1-q^{2(h+j+t-1)}}{1-q^{2% (h+j-1)}}.

ⓘ

Symbols:: $\subseteq$ : is, or is contained in, $h$ : nonnegative integer, $j$ : nonnegative integer, $\pi$ : plane partition, $r$ : positive integer, $t$ : positive integer, $B(r,s,t)$ : Box and $q$ : parameter
Referenced by:: §26.12(ii)
Permalink:: http://dlmf.nist.gov/26.12.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(ii), §26.12 and Ch.26

►

26.12.23

\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ cyclically symmetric}\end{subarray}}q^{\left|\pi\right|}% =\prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h<j\leq r}\frac{1-q^% {3(h+2j-1)}}{1-q^{3(h+j-1)}}=\prod_{h=1}^{r}\left(\frac{1-q^{3h-1}}{1-q^{3h-2}% }\prod_{j=h}^{r}\frac{1-q^{3(r+h+j-1)}}{1-q^{3(2h+j-1)}}\right).

ⓘ

Symbols:: $\subseteq$ : is, or is contained in, $h$ : nonnegative integer, $j$ : nonnegative integer, $\pi$ : plane partition, $r$ : positive integer, $B(r,s,t)$ : Box and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.12.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(ii), §26.12 and Ch.26

… ►where

\sigma_{2}(j)

is the sum of the squares of the divisors of

j

. …

… ►

J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

►

J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

ⓘ

External Links:: ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt
Referenced by:: §19.35(i).
Encodings:: BibTeX

… ►

H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

… ►

H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

… ►

H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

…

… ►Every permutation is a product of transpositions. A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. … ►Every transposition is the product of adjacent transpositions. If

j<k

, then

{\left(j,k\right)}

is a product of

2k-2j-1

adjacent transpositions: …Every permutation is a product of adjacent transpositions. …

… ►

26.9.4

\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}},

n\geq 0

,

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.5

\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1% +\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},

ⓘ

Symbols:: $p_{\NVar{k}}\left(\NVar{n}\right)$ : total number of partitions of $n$ into at most $k$ parts, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.7

\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{k}q^{n}=1+\sum_{k=1}^{\infty% }\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}.

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$ : number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(ii), §26.9 and Ch.26

… ►

26.9.9

p_{k}\left(n\right)=\frac{1}{n}\sum_{t=1}^{n}p_{k}\left(n-t\right)\sum_{\begin% {subarray}{c}j\mathbin{|}t\\ j\leq k\end{subarray}}j,

ⓘ

Symbols:: $p_{\NVar{k}}\left(\NVar{n}\right)$ : total number of partitions of $n$ into at most $k$ parts, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §26.9(iii)
Permalink:: http://dlmf.nist.gov/26.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §26.9(iii), §26.9 and Ch.26

►where the inner sum is taken over all positive divisors of

t

that are less than or equal to

k

. …

… ►

K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

… ►

R. McD. Dodds and G. Wiechers (1972) Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

ⓘ

Notes:: Single-precision Fortran
External Links:: Document, Code
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

… ►

L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

ⓘ

External Links:: Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §18.17(iii).
Encodings:: BibTeX

…

sums%20of%20products

1—10 of 14 matching pages

1: Bibliography C

2: 26.10 Integer Partitions: Other Restrictions

3: 26.14 Permutations: Order Notation

4: 27.2 Functions

5: 12.10 Uniform Asymptotic Expansions for Large Parameter

6: 26.12 Plane Partitions

7: Bibliography V

8: 26.13 Permutations: Cycle Notation

9: 26.9 Integer Partitions: Restricted Number and Part Size

10: Bibliography D