non-classical Freud-type OP’s

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1: 18.39 Applications in the Physical Sciences

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§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. … ►

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

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Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
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2: 7.20 Mathematical Applications

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§7.20(ii) Cornu’s Spiral

►Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\ge 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. … ►

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Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$, $y=S\left(t\right)$, $t\in [0,\mathrm{\infty })$. Magnify

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3: 31.2 Differential Equations

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§31.2(i) Heun’s Equation

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

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http://dlmf.nist.gov/31.2.E1

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Annotations for §31.2(i), §31.2 and Ch.31

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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§31.2(v) Heun’s Equation Automorphisms

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4: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

5: 18.38 Mathematical Applications

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …Each of these typically require a particular non-classical weight functions and analysis of the corresponding OP’s. … ►

Exceptional OP’s

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Non-Classical Weight Functions

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6: 18.36 Miscellaneous Polynomials

… ►Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)). … ►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … ►

§18.36(v) Non-Classical Laguerre Polynomials $L_n^{(-k)}\left(x\right)$, $k=1,2\mathrm{\dots }$

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. … ►Hermite EOP’s are defined in terms of classical Hermite OP’s. …

7: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

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Annotations for §28.2(i), §28.2 and Ch.28

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§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

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§28.2(iv) Floquet Solutions

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8: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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7.2.5 $$F\left(z\right)={\mathrm{e}}^{-z^2}{\int }_0^z{\mathrm{e}}^{t^2}dt.$$

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Defines:

$F\left(z\right)$: Dawson’s integral

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.1

Permalink:

http://dlmf.nist.gov/7.2.E5

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See also:

Annotations for §7.2(ii), §7.2 and Ch.7

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7.2.8 $$S\left(z\right)={\int }_0^z\sin \left(\frac{1}{2}\pi t^2\right)dt,$$

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Defines:

$S\left(z\right)$: Fresnel integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.2

Referenced by:

§7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.2.E8

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Annotations for §7.2(iii), §7.2 and Ch.7

► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

$\underset{x\to \mathrm{\infty }}{lim}S\left(x\right)=\frac{1}{2}.$

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9: 28.20 Definitions and Basic Properties

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§28.20(i) Modified Mathieu’s Equation

►When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: ►

28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

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Annotations for §28.20(i), §28.20 and Ch.28

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28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

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Annotations for §28.20(i), §28.20 and Ch.28

… ►For $s\in \mathbb{Z}$, …

10: Bibliography K

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A proof of Askey’s conjectured $q$-analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19 (4), pp. 969–986.

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External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.13.

Encodings:

BibTeX

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

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Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.

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External Links:

ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§25.12(ii).

Encodings:

BibTeX

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Tom Koornwinder’s Personal Collection of Maple Procedures

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:

§36.14(i).

Encodings:

BibTeX

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