Frobenius’ identity

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1: 21.7 Riemann Surfaces

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§21.7(ii) Fay’s Trisecant Identity

… ►where again all integration paths are identical for all components. … ►

§21.7(iii) Frobenius’ Identity

… ►Then for all ${𝐳}_j\in {ℂ}^g$, $j=1,2,3,4$, such that ${𝐳}_1+{𝐳}_2+{𝐳}_3+{𝐳}_4=0$, and for all ${𝜶}_j$, ${𝜷}_j$ $\in {ℝ}^g$, such that ${𝜶}_1+{𝜶}_2+{𝜶}_3+{𝜶}_4=0$ and ${𝜷}_1+{𝜷}_2+{𝜷}_3+{𝜷}_4=0$, we have Frobenius’ identity: …

2: 31.18 Methods of Computation

… ►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

3: 2.7 Differential Equations

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§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

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2.7.3 $$Q(\alpha )\equiv \alpha (\alpha -1)+f_0\alpha +g_0=0.$$

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Symbols:

$\equiv$: equals by definition, $f_s$: coefficients, $g_s$: coefficients and $Q(\alpha )$: indicial function

Permalink:

http://dlmf.nist.gov/2.7.E3

Encodings:

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See also:

Annotations for §2.7(i), §2.7 and Ch.2

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4: 31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

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§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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5: 24.10 Arithmetic Properties

… ►where $m\equiv n\cancel{\equiv }0(modp-1)$. …valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer. … ►

24.10.8 $$N_{2n}\equiv 0(mod{p}^{\mathrm{\ell }}),$$

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Symbols:

$\equiv$: modular equivalence, $\mathrm{\ell }$: integer, $n$: integer and $p$: prime

Referenced by:

§24.10(iv)

Permalink:

http://dlmf.nist.gov/24.10.E8

Encodings:

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See also:

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

►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. ►

24.10.9

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Symbols:

$E_n$: Euler numbers, $\equiv$: modular equivalence, $\mathrm{\ell }$: integer, $n$: integer and $p$: prime

Referenced by:

§24.10(iv)

Permalink:

http://dlmf.nist.gov/24.10.E9

Encodings:

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See also:

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

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6: 31.11 Expansions in Series of Hypergeometric Functions

… ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. …The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. …Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►Such series diverge for Fuchs–Frobenius solutions. …

7: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. But $y^s\equiv x^{rs}\equiv x^{1+t\varphi \left(n\right)}\equiv x(modn)$, so $y^s$ is the same as $x$ modulo $n$. …

8: 36.9 Integral Identities

§36.9 Integral Identities

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36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integral identity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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9: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

10: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(ii) Typical Identities of Rank 2

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§22.9(iii) Typical Identities of Rank 3

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