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11: 24.2 Definitions and Generating Functions

… ►

B_{2n+1}=0

… ►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

Table 24.2.4: Euler numbers

E_{n}

► ►►

$n$	$E_{n}$
…

► ►

Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

► ►►

	$k$
…

► ►

Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

► ►►

	$k$
…

►

12: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …

13: 10.75 Tables

… ►

•

Wills et al. (1982) tabulates $j_{0,m}$ , $j_{1,m}$ , $y_{0,m}$ , $y_{1,m}$ for $m=1(1)30$ , 35D.

… ►

•

MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_{0}\left(z\right)-iJ_{1}\left(z\right)$ , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

… ►

•

Zhang and Jin (1996, pp. 296–305) tabulates $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , respectively.)

…

14: 27.2 Functions

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. … ►Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. …Note that

J_{1}\left(n\right)=\phi\left(n\right)

. ►In the following examples,

a_{1},\dots,a_{\nu\left(n\right)}

are the exponents in the factorization of

n

in (27.2.1). … ►Table 27.2.1 lists the first 100 prime numbers

p_{n}

. …

15: 3.11 Approximation Techniques

… ►Beginning with

u_{n+1}=0

u_{n}=c_{n}

, we apply … ►With

b_{0}=1

, the last

q

equations give

b_{1},\dots,b_{q}

as the solution of a system of linear equations. … ►(3.11.29) is a system of

n+1

linear equations for the coefficients

a_{0},a_{1},\dots,a_{n}

. … ►With this choice of

a_{k}

and

f_{j}=f(x_{j})

, the corresponding sum (3.11.32) vanishes. … ►Two are endpoints:

(x_{0},y_{0})

and

(x_{3},y_{3})

; the other points

(x_{1},y_{1})

and

(x_{2},y_{2})

are control points. …

16: 3.7 Ordinary Differential Equations

… ►The path is partitioned at

P+1

points labeled successively

z_{0},z_{1},\dots,z_{P}

, with

z_{0}=a

z_{P}=b

. … ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►The values

\lambda_{k}

are the eigenvalues and the corresponding solutions

w_{k}

of the differential equation are the eigenfunctions. … ►where

h=z_{n+1}-z_{n}

and …

17: 3.2 Linear Algebra

… ►where

u_{j}=c_{j}

j=1,2,\dots,n-1

d_{1}=b_{1}

, and …Forward elimination for solving

\mathbf{A}\mathbf{x}=\mathbf{f}

then becomes

y_{1}=f_{1}

, …and back substitution is

x_{n}=y_{n}/d_{n}

, followed by … ►Define the Lanczos vectors

\mathbf{v}_{j}

and coefficients

\alpha_{j}

and

\beta_{j}

\mathbf{v}_{0}=\boldsymbol{{0}}

, a normalized vector

\mathbf{v}_{1}

(perhaps chosen randomly),

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

, and for

j=1,2,\ldots,n-1

by the recursive scheme … ►Start with

\mathbf{v}_{0}=\boldsymbol{{0}}

, vector

\mathbf{v}_{1}

such that

\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

. …

18: 19.29 Reduction of General Elliptic Integrals

… ►Let …where … ►Next, for

j=1,2

, define

Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}

, and assume both

Q

’s are positive for

y<t<x

. …where …If

Q_{1}(t)=(a_{1}+b_{1}t)(a_{2}+b_{2}t)

, where both linear factors are positive for

y<t<x

, and

Q_{2}(t)=f_{2}+g_{2}t+h_{2}t^{2}

, then (19.29.25) is modified so that …

19: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►This transformation is accelerating if

\{s_{n}\}

is a linearly convergent sequence, i. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

20: 18.2 General Orthogonal Polynomials

… ►The

h_{n}

are defined as: … ►Assume that the

p_{n}(x)

are monic, so

A_{n}=1=a_{n}

. … ►Let

x_{1},\ldots,x_{n}

be the zeros of the OP

p_{n}

, so …The discriminant of

p_{n}

is defined by … ►where

x_{1},x_{2},\ldots,x_{n}

are the zeros of

p_{n}(x)

and …