equiconvergence theorem

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1: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

2: 18.2 General Orthogonal Polynomials

… ►Markov’s theorem states that … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_n(x)=T_n\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►For OP’s with orthogonality measure in $𝒮$ Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. … ►See Szegő (1975, Theorem 7.2). …

3: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

4: 30.10 Series and Integrals

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

5: 30.4 Functions of the First Kind

… ►It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$.

6: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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7: 19.35 Other Applications

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§19.35(i) Mathematical

►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)). …

8: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

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

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

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13.13.12 $${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!x^n}U(a-n,b-n,x),$$ .

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Symbols:

$U(a,b,z)$: 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

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

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9: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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10: 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 $x^{t\varphi \left(n\right)}\equiv 1(modn)$. …