q-binomial theorem

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11—20 of 124 matching pages

11: 30.10 Series and Integrals

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

►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)). …

14: 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 $U\left(a,b,z\right)$

…

15: 10.23 Sums

… ►

§10.23(i) Multiplication Theorem

… ►

§10.23(ii) Addition Theorems

►

Neumann’s Addition Theorem

… ►

Graf’s and Gegenbauer’s Addition Theorems

… ► …

16: 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}

. …

17: 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. …

18: 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 $W_{\kappa,\mu}\left(z\right)$

…

19: 22.18 Mathematical Applications

… ►

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►With the identification

x=\operatorname{sn}\left(z,k\right)

y=\ifrac{\mathrm{d}(\operatorname{sn}\left(z,k\right))}{\mathrm{d}z}

, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

20: Hans Volkmer

… ►His book Multiparameter Eigenvalue Problems and Expansion Theorems was published by Springer as Lecture Notes in Mathematics No. …

q-binomial theorem

11—20 of 124 matching pages

11: 30.10 Series and Integrals

12: 10.44 Sums

§10.44(i) Multiplication Theorem

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

13: 19.35 Other Applications

§19.35(i) Mathematical

14: 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 $U\left(a,b,z\right)$

15: 10.23 Sums

§10.23(i) Multiplication Theorem

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

16: 27.16 Cryptography

17: 14.28 Sums

§14.28(i) Addition Theorem

18: 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 $W_{\kappa,\mu}\left(z\right)$

19: 22.18 Mathematical Applications

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

20: Hans Volkmer

q-binomial theorem

11—20 of 124 matching pages

11: 30.10 Series and Integrals

12: 10.44 Sums

§10.44(i) Multiplication Theorem

§10.44(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

13: 19.35 Other Applications

§19.35(i) Mathematical

14: 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 )

15: 10.23 Sums

§10.23(i) Multiplication Theorem

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

Graf’s and Gegenbauer’s Addition Theorems

16: 27.16 Cryptography

17: 14.28 Sums

§14.28(i) Addition Theorem

18: 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 )

19: 22.18 Mathematical Applications

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

20: Hans Volkmer

§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 $W_{\kappa,\mu}\left(z\right)$