von Staudt–Clausen theorem

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21—30 of 130 matching pages

21: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

23: 16.12 Products

… ►The following formula is often referred to as Clausen’s formula …

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute

\pi

to high precision (Borwein and Borwein (1987, p. 26)). …

25: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

Watson’s $q$ -Analog of Whipple’s Theorem

… ►

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)

…

26: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

►

§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

… ►

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

… ►

13.13.12

e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),

|y|<|x|

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §13.13(ii), §13.13 and Ch.13

►

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

…

27: 10.23 Sums

… ►

§10.23(i) Multiplication Theorem

… ►

§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

… ►

Graf’s and Gegenbauer’s Addition Theorems

… ► …

28: 27.16 Cryptography

… ►Thus,

y\equiv x^{r}\pmod{n}

and

1\leq y<n

. … ►By the Euler–Fermat theorem (27.2.8),

x^{\phi\left(n\right)}\equiv 1\pmod{n}

; hence

x^{t\phi\left(n\right)}\equiv 1\pmod{n}

. …

29: 14.28 Sums

… ►

§14.28(i) Addition Theorem

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …

30: 13.26 Addition and Multiplication Theorems

§13.26 Addition and Multiplication Theorems

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§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

… ►

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

… ►

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$

…

von Staudt–Clausen theorem

21—30 of 130 matching pages

21: 30.10 Series and Integrals

22: 10.44 Sums

§10.44(i) Multiplication Theorem

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

23: 16.12 Products

24: 19.35 Other Applications

§19.35(i) Mathematical

25: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

Watson’s $q$ -Analog of Whipple’s Theorem

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)

26: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

27: 10.23 Sums

§10.23(i) Multiplication Theorem

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

28: 27.16 Cryptography

29: 14.28 Sums

§14.28(i) Addition Theorem

30: 13.26 Addition and Multiplication Theorems

§13.26 Addition and Multiplication Theorems

§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$

von Staudt–Clausen theorem

21—30 of 130 matching pages

21: 30.10 Series and Integrals

22: 10.44 Sums

§10.44(i) Multiplication Theorem

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

23: 16.12 Products

24: 19.35 Other Applications

§19.35(i) Mathematical

25: 17.9 Further Transformations of ϕ r r + 1 Functions

Watson’s q -Analog of Whipple’s Theorem

Gasper’s q -Analog of Clausen’s Formula (16.12.2)

26: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

§13.13(i) Addition Theorems for M ⁡ ( a , b , z )

§13.13(ii) Addition Theorems for U ⁡ ( a , b , z )

§13.13(iii) Multiplication Theorems for M ⁡ ( a , b , z ) and U ⁡ ( a , b , z )

27: 10.23 Sums

§10.23(i) Multiplication Theorem

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

28: 27.16 Cryptography

29: 14.28 Sums

§14.28(i) Addition Theorem

30: 13.26 Addition and Multiplication Theorems

§13.26 Addition and Multiplication Theorems

§13.26(i) Addition Theorems for M κ , μ ⁡ ( z )

§13.26(ii) Addition Theorems for W κ , μ ⁡ ( z )

§13.26(iii) Multiplication Theorems for M κ , μ ⁡ ( z ) and W κ , μ ⁡ ( z )

25: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

Watson’s $q$ -Analog of Whipple’s Theorem

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)

§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$