relation%20to%20minimax%20polynomials

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11: 25.12 Polylogarithms

… ►The remainder of the equations in this subsection apply to principal branches. … ►The special case

z=1

is the Riemann zeta function:

\zeta\left(s\right)=\operatorname{Li}_{s}\left(1\right)

. … ►Further properties include …and … ►In terms of polylogarithms …

Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$ , $xe^{x}E_{1}\left(x\right)$ , and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ . These are included in Abramowitz and Stegun (1964, Ch. 5).

►

•

Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

►

•

Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

►

•

MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

… ►

•

Luke (1969b, pp. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

…

14: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

15: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … Ford Award, given to recognize authors of articles of expository excellence. … ►

25 Zeta and Related Functions

…

16: 26.3 Lattice Paths: Binomial Coefficients

… ►

§26.3(i) Definitions

►

\genfrac{(}{)}{0.0pt}{}{m}{n}

is the number of ways of choosing

n

objects from a collection of

m

distinct objects without regard to order.

\genfrac{(}{)}{0.0pt}{}{m+n}{n}

is the number of lattice paths from

(0,0)

(m,n)

. …The number of lattice paths from

(0,0)

(m,n)

m\leq n

, that stay on or above the line

y=x

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

§26.3(iii) Recurrence Relations

…

17: 27.2 Functions

… ►(

\nu\left(1\right)

is defined to be 0.) Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the number of

k

-tuples of integers

\leq n

whose greatest common divisor is relatively prime to

n

. …

18: 26.5 Lattice Paths: Catalan Numbers

… ►

§26.5(i) Definitions

… ►It counts the number of lattice paths from

(0,0)

(n,n)

that stay on or above the line

y=x

. … ►

§26.5(iii) Recurrence Relations

… ►

26.5.6

C\left(n\right)\sim\frac{4^{n}}{\sqrt{\pi n^{3}}},

n\to\infty

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

19: Bibliography I

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions,

q

-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

ⓘ

External Links:: ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

►

M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

… ►

A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

ⓘ

Notes:: In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207
External Links:: ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

…

20: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

§26.4(i) Definitions

… ►It is also the number of

k

-dimensional lattice paths from

(0,0,\ldots,0)

(n_{1},n_{2},\ldots,n_{k})

. For

k=0,1

, the multinomial coefficient is defined to be

1

. … ►(The empty set is considered to have one permutation consisting of no cycles.) … ►

§26.4(iii) Recurrence Relation

…