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1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

â–ºThe origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … â–º

Euler Numbers and Polynomials

… â–ºIts coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 19.11 Addition Theorems

… â–º

\Delta(\theta)=\sqrt{1-k^{2}{\sin}^{2}\theta}.

… â–º

\delta=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).

… â–º

19.11.6_5

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).

â“˜

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
Permalink:: http://dlmf.nist.gov/19.11.E6_5
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.28):: This equation was given a number.
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… â–º

\delta=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).

… â–ºIf

\phi=\theta

in §19.11(i) and

\Delta(\theta)

is again defined by (19.11.3), then …

3: 31.2 Differential Equations

… â–ºThis equation has regular singularities at

0,1,a,\infty

, with corresponding exponents

\{0,1-\gamma\}

\{0,1-\delta\}

\{0,1-\epsilon\}

\{\alpha,\beta\}

, respectively (§2.7(i)). … â–ºThe parameters play different roles:

a

is the singularity parameter;

\alpha,\beta,\gamma,\delta,\epsilon

are exponent parameters;

q

is the accessory parameter. … â–ºNext,

w(z)=(z-1)^{1-\delta}w_{2}(z)

satisfies (31.2.1) if

w_{2}

is a solution of (31.2.1) with transformed parameters

q_{2}=q+a\gamma(1-\delta)

;

\alpha_{2}=\alpha+1-\delta

\beta_{2}=\beta+1-\delta

\delta_{2}=2-\delta

. … â–ºFor example, if

\tilde{z}=z/a

, then the parameters are

\tilde{a}=1/a

\tilde{q}=q/a

;

\tilde{\delta}=\epsilon

\tilde{\epsilon}=\delta

. …For example,

w(z)=(1-z)^{-\alpha}\tilde{w}(z/(z-1))

, which arises from

\tilde{z}=z/(z-1)

, satisfies (31.2.1) if

\tilde{w}(\tilde{z})

is a solution of (31.2.1) with

z

replaced by

\tilde{z}

and transformed parameters

\tilde{a}=a/(a-1)

\tilde{q}=-(q-a\alpha\gamma)/(a-1)

;

\tilde{\beta}=\alpha+1-\delta

\tilde{\delta}=\alpha+1-\beta

. …

4: 13.27 Mathematical Applications

… â–º

13.27.1

g=\begin{pmatrix}1&\alpha&\beta\\ 0&\gamma&\delta\\ 0&0&1\end{pmatrix},

â“˜

Permalink:: http://dlmf.nist.gov/13.27.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.27 and Ch.13

â–ºwhere

\alpha

\beta

\gamma

\delta

are real numbers, and

\gamma>0

. …

5: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

â–º

§1.17(i) Delta Sequences

… â–º

Sine and Cosine Functions

… â–º

Coulomb Functions (§33.14(iv))

… â–º

Airy Functions (§9.2)

…

6: 14.16 Zeros

… â–ºwhere

m

n\in\mathbb{Z}

and

\delta_{\mu}

\delta_{\nu}\in(0,1)

. … â–ºThe number of zeros of

\mathsf{P}^{\mu}_{\nu}\left(x\right)

in the interval

(-1,1)

\max(\lceil\nu-|\mu|\rceil,0)

if any of the following sets of conditions hold: … â–º

(b)

$\mu>0$ , $n\geq m$ , and $\delta_{\nu}>\delta_{\mu}$ .

… â–ºThe number of zeros of

\mathsf{P}^{\mu}_{\nu}\left(x\right)

in the interval

(-1,1)

\max(\lceil\nu-|\mu|\rceil,0)+1

if either of the following sets of conditions holds: â–º

(a)

$\mu>0$ , $n>m$ , and $\delta_{\nu}\leq\delta_{\mu}$ .

…

7: 3.9 Acceleration of Convergence

… â–ºHere

\Delta

is the forward difference operator: … â–º

§3.9(iii) Aitken’s $\Delta^{2}$ -Process

… â–º … â–ºShanks’ transformation is a generalization of Aitken’s

\Delta^{2}

-process. … â–ºAitken’s

\Delta^{2}

-process is the case

k=1

. …

8: 27.13 Functions

… â–º

§27.13(i) Introduction

… â–ºThe subsections that follow describe problems from additive number theory. … â–º

§27.13(ii) Goldbach Conjecture

… â–º

§27.13(iii) Waring’s Problem

… â–ºwhere

\delta_{1}(n)

and

\delta_{3}(n)

are the number of divisors of

n

congruent respectively to 1 and 3 (mod 4), and by equating coefficients in (27.13.5) and (27.13.6) Jacobi deduced that …

9: 8.11 Asymptotic Approximations and Expansions

… â–ºwhere

\delta

denotes an arbitrary small positive constant. … â–ºThis expansion is absolutely convergent for all finite

z

, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of

\gamma\left(a,z\right)

a\to\infty

|\operatorname{ph}a|\leq\pi-\delta

. … â–ºThe expansion (8.11.7) also applies when

a

is replaced by

-a

\lambda<0

and

|{\rm ph}\,a|\leq\frac{3\pi}{2}-\omega-\delta

with

\omega={\rm ph}(-\lambda+\ln\left(-\lambda\right)+\pi i)

0<\omega<\pi

. … â–ºuniformly for

x\in(-\infty,1-\delta]

, with … â–ºFor a uniformly valid expansion for

n\to\infty

and

x\in[\delta,1]

, see Wong (1973b). …

10: 18.25 Wilson Class: Definitions

… â–ºFor the Wilson class OP’s

p_{n}(x)

with

x=\lambda(y)

: if the

y

-orthogonality set is

\{0,1,\dots,N\}

, then the role of the differentiation operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the operator

\Delta_{y}

followed by division by

\Delta_{y}(\lambda(y))

, or by the operator

\nabla_{y}

followed by division by

\nabla_{y}(\lambda(y))

. Alternatively if the

y

-orthogonality interval is

(0,\infty)

, then the role of

\ifrac{\mathrm{d}}{\mathrm{d}x}

is played by the operator

\delta_{y}

followed by division by

\delta_{y}(\lambda(y))

. … â–ºTable 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials

W_{n}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

. … â–º

\gamma,\delta>-1,\quad\beta<-N-\delta.

… â–ºThe first four sets imply

\gamma+\delta>-2

, and the last four imply

\gamma+\delta<-2N

. …

delta airlines 716 351 6210 reservations contact number

1—10 of 367 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

2: 19.11 Addition Theorems

3: 31.2 Differential Equations

4: 13.27 Mathematical Applications

5: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

§1.17(i) Delta Sequences

Sine and Cosine Functions

Coulomb Functions (§33.14(iv))

Airy Functions (§9.2)

6: 14.16 Zeros

7: 3.9 Acceleration of Convergence

§3.9(iii) Aitken’s Î” 2 -Process

8: 27.13 Functions

§27.13(i) Introduction

§27.13(ii) Goldbach Conjecture

§27.13(iii) Waring’s Problem

9: 8.11 Asymptotic Approximations and Expansions

10: 18.25 Wilson Class: Definitions

§3.9(iii) Aitken’s $\Delta^{2}$ -Process