Christoffel–Darboux formula

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21: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

22: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). ►The function

w=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+y,x-y\right)

satisfies an Euler–Poisson–Darboux equation: …

23: 25.4 Reflection Formulas

§25.4 Reflection Formulas

… ►

25.4.1

\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (12), p. 259)
Referenced by:: §25.10(i), §5.17
Permalink:: http://dlmf.nist.gov/25.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

►

25.4.2

\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (13), p. 259)
A&S Ref:: 23.2.6
Permalink:: http://dlmf.nist.gov/25.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

… ►

25.4.3

\xi\left(s\right)=\xi\left(1-s\right),

ⓘ

Symbols:: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, p. 260)
Permalink:: http://dlmf.nist.gov/25.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

…

24: 2.2 Transcendental Equations

… ►where

F_{0}=f_{0}

and

sF_{s}

(

s\geq 1

) is the coefficient of

x^{-1}

in the asymptotic expansion of

(f(x))^{s}

(Lagrange’s formula for the reversion of series). …

25: 25.19 Tables

… ►

•

Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ , $x=-5(.05)25$ , to 12S.

►

•

Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$ . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$ , and §22.17 lists tables for some Dirichlet $L$ -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

26: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►§29.15(i) includes formulas for normalizing the eigenvectors. …

27: 14.28 Sums

… ►

§14.28(ii) Heine’s Formula

…

28: Need Help?

… ►

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29: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …

30: 12.16 Mathematical Applications

… ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …