Dougall very well-poised sum

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11: Diego Dominici

… ►Diego was very active in the SIAM activity group on Orthogonal Polynomials and Special Functions since 2010. …

12: 27.13 Functions

… ►The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. … ►This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. …

13: About MathML

… ►That format displays mathematics as static images and is neither very scalable nor accessible, but serves as a workable fallback. …

14: Foreword

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

15: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►These recursions can be used to calculate $p\left(n\right)$, which grows very rapidly. …and $s(h,k)$ is a Dedekind sum given by … ►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by …where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). …

16: 27.16 Cryptography

… ►Procedures for finding such primes require very little computer time. …

17: 3.11 Approximation Techniques

… ►as well as an orthogonal property with respect to sums, as follows. … ►Furthermore, if $f\in C^{\mathrm{\infty }}[-1,1]$, then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with $k$ arbitrary. … ►Because the series (3.11.12) converges rapidly we obtain a very good first approximation to the minimax polynomial $p_n(x)$ for $[a,b]$ if we truncate (3.11.12) at its $(n+1)$th term. … ►Then the sum of the truncated expansion equals $\frac{1}{2}(u_0-u_2)$. … ►With this choice of $a_k$ and $f_j=f(x_j)$, the corresponding sum (3.11.32) vanishes. …

18: 2.11 Remainder Terms; Stokes Phenomenon

… ►In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … ►Multiplying these differences by ${(-1)}^j{2}^{-j-1}$ and summing, we obtain …Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. … ►The next column lists the partial sums $s_n=a_0+a_1+\mathrm{\cdots }+a_n$. …

19: 3.4 Differentiation

… ►

3.4.1 $$h{f}_t^{\prime }=h{f}^{\prime }(x_0+th)=\sum _{k=n_0}^{n_1}{B}_k^n{f}_k+h{R}_{n,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4 and Ch.3

… ►

3.4.7 $$h{f}_t^{\prime }=\sum _{k=-1}^2{B}_k^3{f}_k+h{R}_{3,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

A&S Ref:

25.3.5 (modification of)

Permalink:

http://dlmf.nist.gov/3.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►

3.4.11 $$h{f}_t^{\prime }=\sum _{k=-2}^3{B}_k^5{f}_k+h{R}_{5,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

Permalink:

http://dlmf.nist.gov/3.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►

3.4.15 $$h{f}_t^{\prime }=\sum _{k=-3}^4{B}_k^7{f}_k+h{R}_{7,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

Permalink:

http://dlmf.nist.gov/3.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

20: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).