formally self-adjoint differential operators

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11: 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. … ►However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

…

12: 1.17 Integral and Series Representations of the Dirac Delta

… ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): …Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): … ►By analogy with §1.17(ii) we have the formal series representation …

13: 2.9 Difference Equations

… ►in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). … ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). Formal solutions are … ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►The coefficients $b_s$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha }_2-{\alpha }_1=0$, or with $b_0=1$ when ${\alpha }_2-{\alpha }_1=1,2,3,\mathrm{\dots }$. …

14: 3.10 Continued Fractions

… ►Every convergent, asymptotic, or formal series … ►We say that it corresponds to the formal power series … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►($\nabla$ is the backward difference operator.) …

15: 2.3 Integrals of a Real Variable

… ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►

If $p(b)=\mathrm{\infty }$, then $P_0(b)=0$ and each of the integrals

2.3.22 $$\int {\mathrm{e}}^{\mathrm{i}xp(t)}{P}_s(t)p^{\prime }(t)dt,$$ $s=0,1,2,\mathrm{\dots }$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $p(t)$: real function and $P_s(t)$: function

Permalink:

http://dlmf.nist.gov/2.3.E22

Encodings:

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See also:

Annotations for §2.3(iv), §2.3 and Ch.2

converges at $t=b$ uniformly for all sufficiently large $x$.

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. … ►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …

16: 18.2 General Orthogonal Polynomials

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§18.2(ii) $x$-Difference Operators

… ►It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. … ►where $f(t)$ and $u(t)$ are formal power series in $t$, with $f(0)=1$, $u(0)=0$ and $u^{\prime }(0)=1$. …If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with $c_n=1$ is that … …

17: 16.11 Asymptotic Expansions

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§16.11(i) Formal Series

►For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows: ►

16.11.1 $$E_{p,q}(z)={(2\pi )}^{(p-q)/2}{\kappa }^{-\nu -(1/2)}{\mathrm{e}}^{\kappa z^{1/\kappa }}\sum _{k=0}^{\mathrm{\infty }}{c}_k{\left(\kappa z^{1/\kappa }\right)}^{\nu -k},$$ ,

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Defines:

$E_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\kappa$ and $\nu$

Referenced by:

§16.11(i), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E1

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Annotations for §16.11(i), §16.11 and Ch.16

►

16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

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Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

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See also:

Annotations for §16.11(i), §16.11 and Ch.16

… ►The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and …

18: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … ►

$a_n={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos \left(nx\right)dx,$ $n=0,1,2,\mathrm{\dots }$,

… ►

1.8.4 $$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x){\mathrm{e}}^{-\mathrm{i}nx}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.17(iii), §1.18(v), §1.8(i)

Permalink:

http://dlmf.nist.gov/1.8.E4

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Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.10 $${\int }_a^{b}f(x){\mathrm{e}}^{\mathrm{i}\lambda x}dx\to 0,$$ as $\lambda \to \mathrm{\infty }$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\lambda$: real

Referenced by:

§1.8(i)

Permalink:

http://dlmf.nist.gov/1.8.E10

Encodings:

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See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.15 $$\frac{1}{2}f(0)+\sum _{n=1}^{\mathrm{\infty }}f(n)={\int }_0^{\mathrm{\infty }}f(x)dx+2\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}f(x)\cos \left(2\pi nx\right)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.8.E15

Encodings:

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See also:

Annotations for §1.8(iv), §1.8 and Ch.1

…

19: 2.7 Differential Equations

§2.7 Differential Equations

… ►Formal solutions are … ►provided that . …and $𝒱$ denotes the variational operator (§2.3(i)). … ►Assuming also , we have …

20: Bibliography K

… ►

Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

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Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides. Akademie Verlag, Berlin-New York.

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External Links:

ISBN 3-05-501598-3, MathReview (Apostolos Vourdas), ZentralBlatt

Referenced by:

§10.73(i).

Encodings:

BibTeX

… ►

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

Encodings:

BibTeX