which are also Sheffer polynomials

(0.003 seconds)

11—20 of 677 matching pages

11: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters

\alpha

and

\beta

are fixed and the ratio

n/N=c

is a constant in the interval (0,1), uniform asymptotic formulas (as

n\to\infty

) of the Hahn polynomials

Q_{n}(z;\alpha,\beta,N)

can be found in Lin and Wong (2013) for

z

in three overlapping regions, which together cover the entire complex plane. … ►Asymptotic approximations are also provided for the zeros of

K_{n}\left(x;p,N\right)

in various cases depending on the values of

p

and

\mu

. … ►Taken together, these expansions are uniformly valid for

-\infty<x<\infty

and for

a

in unbounded intervals—each of which contains

[0,(1-\delta)n]

, where

\delta

again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998). … ►Compare also (18.21.12). …

… ►²² 2 D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology, CRC Press, 2001. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. … ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

14: 35.12 Software

… ►Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. … ►

•

Demmel and Koev (2006). Computation of zonal polynomials in MATLAB.

… ►

•

Stembridge (1995). Maple software for zonal polynomials.

►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

15: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►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}

. … ►

•

Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

►

•

Buhler et al. (1992) uses the expansion

24.19.3

\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii)
Permalink:: http://dlmf.nist.gov/24.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

►

•

A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$ . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

16: Bille C. Carlson

… ►Navy after the onset of World War II and worked on the island of Guam with radar, which was novel at the time. … ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

17: 18.38 Mathematical Applications

… ►See also the paragraph on DVRs, below. … ►See also §14.30. … ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. …Algebraic structures were built of which special representations involve Dunkl type operators. … ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …

18: 26.20 Physical Applications

… ►The latter reference also describes chemical applications of other combinatorial techniques. ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

19: 26.22 Software

… ►Also included are websites operated by university departments and consortia, research institutions, and peer-reviewed journals. Citations in the bulleted list refer to sources from which research software can be found and downloaded via the Web. …

20: 34.15 Software

… ►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web. … ►See also Lai and Chiu (1990) and Srinivasa Rao and Rajeswari (1993, Appendix C, pp. 266–292).