Euler pentagonal number theorem

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31: 19.11 Addition Theorems

§19.11 Addition Theorems

… ►

$$\gamma =(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2)(({\csc }^2\psi )-{\alpha }^2),$$

… ►

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

… ►

$$\gamma =(1-{\alpha }^2)(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2),$$

… ►

$$\gamma ={(({\csc }^2\theta )-{\alpha }^2)}^2(({\csc }^2\psi )-{\alpha }^2),$$

…

32: 25.15 Dirichlet $L$-functions

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). … ►

25.15.5 $$L(1-s,\chi )=\frac{k^{s-1}\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}+\chi (-1){\mathrm{e}}^{\pi \mathrm{i}s/2}\right)G(\chi )L(s,\overline{\chi }),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

connection formula

Source:

Apostol (1976, (16), p. 263)

Referenced by:

§25.15(ii)

Permalink:

http://dlmf.nist.gov/25.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are: ►

25.15.10

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

33: 5.5 Functional Relations

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5.5.1 $$\mathrm{\Gamma }\left(z+1\right)=z\mathrm{\Gamma }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.1.15

Referenced by:

§15.4(ii), §15.4(iii), (25.11.27), (25.5.6), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(i), §5.5 and Ch.5

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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5.5.5 $$\mathrm{\Gamma }\left(2z\right)={\pi }^{-1/2}{2}^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

6.1.18

Referenced by:

§14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

… ►

§5.5(iv) Bohr–Mollerup Theorem

►If a positive function $f(x)$ on $(0,\mathrm{\infty })$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\mathrm{\Gamma }\left(x\right)$.

34: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by …The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. With $a_1=0$ and $a_2=1$ the total number of free parameters is $3N-3$. … ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…

35: 27.21 Tables

§27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. Table II lists all solutions $n$ of the equation $f\left(n\right)=m$ for all $m\le 2500$, where $f\left(n\right)$ is defined by (27.14.2). …6 lists $\varphi \left(n\right),d\left(n\right)$, and $\sigma (n)$ for $n\le 1000$; Table 24. … ►

36: 25.2 Definition and Expansions

… ►where the Stieltjes constants ${\gamma }_n$ are defined via … ►

§25.2(iii) Representations by the Euler–Maclaurin Formula

►

25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

… ►For $B_{2k}$ see §24.2(i), and for ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).

37: 25.1 Special Notation

… ► ►►►►►►

$k,m,n$	nonnegative integers.
$p$	prime number.
…
$\gamma$	Euler’s constant (§5.2(ii)).
$\psi \left(x\right)$	digamma function ${\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ except in §25.16. See §5.2(i).
$B_n,B_n\left(x\right)$	Bernoulli number and polynomial (§24.2(i)).
…

…

38: 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:

pMML, png, TeX

See also:

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

… ►

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:

pMML, png, TeX

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). …

39: Karl Dilcher

… ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. … ►

24 Bernoulli and Euler Polynomials

…

40: 25.10 Zeros

… ►Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). … ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

►is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. … ►Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta \left(s\right)$ in that portion of the critical strip with . By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\zeta \left(s\right)$ has any zeros off the line in this region. …