Euler%E2%80%93Fermat%20theorem

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11: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

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Note on Dr. Vacca’s series for $\gamma$ . Quart. J. Math. 43, pp. 215–216.

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External Links:

ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(ii).

Encodings:

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:

MathReview (M. Marden), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

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12: 25.16 Mathematical Applications

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§25.16(ii) Euler Sums

►Euler sums have the form … ► $H\left(s\right)$ is the special case $H(s,1)$ of the function …which satisfies the reciprocity law …when both $H(s,z)$ and $H(z,s)$ are finite. …

13: 5.4 Special Values and Extrema

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$\mathrm{\Gamma }\left(1\right)=1,$

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$n!=\mathrm{\Gamma }\left(n+1\right).$

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5.4.11 $${\mathrm{\Gamma }}^{\prime }\left(1\right)=-\gamma .$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\gamma$: Euler’s constant

A&S Ref:

6.3.2

Referenced by:

§5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E11

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See also:

Annotations for §5.4(i), §5.4 and Ch.5

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$\psi \left(1\right)=-\gamma ,$

… ►For error bounds for this estimate see Walker (2007, Theorem 5).

14: 5.18 $q$-Gamma and $q$-Beta Functions

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5.18.5 $${\mathrm{\Gamma }}_q\left(1\right)={\mathrm{\Gamma }}_q\left(2\right)=1,$$

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function and $q$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E5

Encodings:

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See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.7 $${\mathrm{\Gamma }}_q\left(z+1\right)=\frac{1-q^z}{1-q}{\mathrm{\Gamma }}_q\left(z\right).$$

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.18.E7

Encodings:

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See also:

Annotations for §5.18(ii), §5.18 and Ch.5

►Also, $\ln {\mathrm{\Gamma }}_q\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. … ►For generalized asymptotic expansions of $\ln {\mathrm{\Gamma }}_q\left(z\right)$ as $|z|\to \mathrm{\infty }$ see Olde Daalhuis (1994) and Moak (1984). For the $q$-digamma or $q$-psi function ${\psi }_q(z)={\mathrm{\Gamma }}_q^{\prime }\left(z\right)/{\mathrm{\Gamma }}_q\left(z\right)$ see Salem (2013). …

15: Bibliography C

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A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

Encodings:

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$\gamma$ rays from an extranuclear direct capture process. Nuclear Physics 24 (1), pp. 89–101.

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External Links:

Document

Referenced by:

§33.22(v).

Encodings:

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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External Links:

Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

Encodings:

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

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16: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

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§24.10(ii) Kummer Congruences

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24.10.5 $$E_n\equiv E_{n+p-1}(modp),$$

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Symbols:

$E_n$: Euler numbers, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.10(ii)

Permalink:

http://dlmf.nist.gov/24.10.E5

Encodings:

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See also:

Annotations for §24.10(ii), §24.10 and Ch.24

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§24.10(iv) Factors

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17: 15.10 Hypergeometric Differential Equation

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: ►

15.10.17 $$w_3(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(b-c+1\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_2(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_3(z)$: Kummer’s solution $w_3$

Permalink:

http://dlmf.nist.gov/15.10.E17

Encodings:

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See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.18 $$w_4(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1-b\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_2(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_4(z)$: Kummer’s solution $w_4$

Permalink:

http://dlmf.nist.gov/15.10.E18

Encodings:

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See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.21 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_4(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_4(z)$: Kummer’s solution $w_4$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E21

Encodings:

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See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.25 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(b-a\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}{w}_5(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_6(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_5(z)$: Kummer’s solution $w_5$ and $w_6(z)$: Kummer’s solution $w_6$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E25

Encodings:

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See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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18: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

►As ${\gamma }^2\to +\mathrm{\infty }$, with $q=2(n-m)+1$, … ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982). …

19: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

20: 32.8 Rational Solutions

… ►In the general case assume $\gamma \delta \ne 0$, so that as in §32.2(ii) we may set $\gamma =1$ and $\delta =-1$. … ►

$\alpha =\frac{1}{2}{(m+\epsilon \gamma )}^2$ and $\beta =-\frac{1}{2}{n}^2$, where $n>0$, $m+n$ is odd, and $\alpha \ne 0$ when .

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$\alpha =\frac{1}{2}{a}^2$, $\beta =-\frac{1}{2}{(a+n)}^2$, and $\gamma =m$, with $m+n$ even.

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$\alpha =\frac{1}{2}{(b+n)}^2$, $\beta =-\frac{1}{2}{b}^2$, and $\gamma =m$, with $m+n$ even.

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$\alpha =\frac{1}{8}{(2m+1)}^2$, $\beta =-\frac{1}{8}{(2n+1)}^2$, and $\gamma \notin \mathbb{Z}$.

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