Picard theorem

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31: 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). … ►Part of this theorem was already proved by Blumenthal (1898). … ►See Szegő (1975, Theorem 7.2). …

32: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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$q$-Binomial Theorem

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33: 18.18 Sums

… ►See Szegő (1975, Theorems 3.1.5 and 5.7.1). … ►

§18.18(ii) Addition Theorems

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Ultraspherical

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Legendre

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§18.18(iii) Multiplication Theorems

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34: 25.15 Dirichlet $L$-functions

… ►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

ⓘ

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

35: Tom M. Apostol

… ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …

36: Bille C. Carlson

… ►In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. …

37: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.23 $${\mathrm{\Phi }}_{n+1}(z)=z{\mathrm{\Phi }}_n(z)-\overline{{\alpha }_n}{\mathrm{\Phi }}_n^{\ast }(z),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

►

18.33.24 $${\mathrm{\Phi }}_{n+1}^{\ast }(z)={\mathrm{\Phi }}_n^{\ast }(z)-{\alpha }_{n}z{\mathrm{\Phi }}_n(z).$$

ⓘ

Symbols:

$z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

… ►

Verblunsky’s Theorem

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Szegő’s Theorem

►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …

38: 10.60 Sums

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§10.60(i) Addition Theorems

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39: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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§28.5(i) Definitions

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Theorem of Ince (1922)

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40: 28.29 Definitions and Basic Properties

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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