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11: Bibliography B

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

Encodings:

BibTeX

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Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

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External Links:

FullText

Referenced by:

2nd item.

Encodings:

BibTeX

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Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (2), pp. 575–580.

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Notes:

Erratum: same journal v. 48 (1975), p. 518

External Links:

FullText, MathReview (H. A. Lauwerier), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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A note on Mathieu functions. Proc. Nederl. Akad. Wetensch. 51 (7), pp. 891–893=Indagationes Math. 10, 319–321 (1948).

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External Links:

MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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A Course in Computational Number Theory. Key College Publishing, Emeryville, CA.

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Notes:

With 1 CD-ROM (Windows, Macintosh and UNIX) containing a collection of Mathematica programs for computations in number theory.

External Links:

ISBN 1-930190-10-7, MathReview (Samuel S. Wagstaff, Jr.), ZentralBlatt

Referenced by:

§27.12, §27.19, §27.21, 2nd item, Profile David M. Bressoud.

Encodings:

BibTeX

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12: 19.11 Addition Theorems

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

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

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: 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

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

… ►If $\varphi =\theta$ in §19.11(i) and $\mathrm{\Delta }(\theta )$ is again defined by (19.11.3), then …

13: 31.2 Differential Equations

… ►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ►Next, $w(z)={(z-1)}^{1-\delta }{w}_2(z)$ satisfies (31.2.1) if $w_2$ is a solution of (31.2.1) with transformed parameters $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. … ►For example, if $\tilde{z}=z/a$, then the parameters are $\tilde{a}=1/a$, $\tilde{q}=q/a$; $\tilde{\delta }=ϵ$, $\widetilde{ϵ}=\delta$. …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 $\tilde{a}=a/(a-1)$, $\tilde{q}=-(q-a\alpha \gamma )/(a-1)$; $\tilde{\beta }=\alpha +1-\delta$, $\tilde{\delta }=\alpha +1-\beta$. …

14: 13.27 Mathematical Applications

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13.27.1

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Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. …

15: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. … ►

26.11.2 $$c_m\left(0\right)={\delta }_{0,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

16: 14.16 Zeros

… ►where $m$, $n\in \mathbb{Z}$ and ${\delta }_{\mu }$, ${\delta }_{\nu }\in (0,1)$. … ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold: … ►

$\mu >0$, $n\ge m$, and ${\delta }_{\nu }>{\delta }_{\mu }$.

… ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds: ►

$\mu >0$, $n>m$, and ${\delta }_{\nu }\le {\delta }_{\mu }$.

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17: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(i) Delta Sequences

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Sine and Cosine Functions

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Coulomb Functions (§33.14(iv))

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Airy Functions (§9.2)

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18: 27.13 Functions

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§27.13(i) Introduction

… ►The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

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§27.13(iii) Waring’s Problem

… ►where ${\delta }_1(n)$ 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 …

19: 3.9 Acceleration of Convergence

… ►Here $\mathrm{\Delta }$ is the forward difference operator: … ►

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

… ► … ►Shanks’ transformation is a generalization of Aitken’s ${\mathrm{\Delta }}^2$-process. … ►Aitken’s ${\mathrm{\Delta }}^2$-process is the case $k=1$. …

20: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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