Stieltjes%E2%80%93Wigert%20polynomials

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11: Bibliography

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A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.

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External Links:: ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §31.15(ii).
Encodings:: BibTeX

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M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:: ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt
Referenced by:: §31.15(ii).
Encodings:: BibTeX

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D. E. Amos, S. L. Daniel, and M. K. Weston (1977) Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions

I_{\nu}(x)

and

J_{\nu}(x)

x\geq 0

\nu\geq 0

. ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:: Single-precision Fortran, maximum accuracy 16S.
External Links:: ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

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D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:: Double- and single-precision Fortran, maximum accuracy 18S or 14S.
External Links:: ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

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M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt
Referenced by:: §18.15(vi), §18.15(vi).
Encodings:: BibTeX

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12: Bibliography H

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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T. H. Hildebrandt (1938) Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:: ISSN 0002-9890, Document, Link, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

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13: Bibliography J

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M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:: ISSN 0167-2789, Document, MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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W. B. Jones and W. Van Assche (1998) Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:: MathReview (Olav Njåstad), ZentralBlatt
Referenced by:: §5.10, §5.10.
Encodings:: BibTeX

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B. R. Judd (1976) Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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External Links:: ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)
Referenced by:: §34.12.
Encodings:: BibTeX

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14: Bibliography R

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M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

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W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

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W. P. Reinhardt (2021b) Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

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15: Bibliography K

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K. W. J. Kadell (1994) A proof of the

q

-Macdonald-Morris conjecture for

BC_{n}

. Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:: ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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T. H. Koornwinder (2015) 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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16: 25.6 Integer Arguments

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§25.6(i) Function Values

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25.6.3

\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1},

n=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $n$ : nonnegative integer
Source:: Apostol (1976, (20), p. 266); with $n\neq 0$
A&S Ref:: 23.2.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/25.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

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25.6.6

\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{% 2k+1}\left(t\right)\cot\left(\pi t\right)\,\mathrm{d}t,

k=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral and $k$ : nonnegative integer
Source:: Nörlund (1924, (81*), p. 66); with $\nu=k$
A&S Ref:: 23.2.17
Permalink:: http://dlmf.nist.gov/25.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

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25.6.12

\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Apostol (1985a, pp. 226, 227, 229)
Referenced by:: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

►where

\gamma_{1}

is given by (25.2.5). …

17: 18.1 Notation

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Classical OP’s

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Hahn Class OP’s

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Wilson Class OP’s

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Stieltjes–Wigert: $S_{n}\left(x;q\right)$ .

… ►Nor do we consider the shifted Jacobi polynomials: …

18: Bibliography W

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X.-S. Wang and R. Wong (2011) Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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External Links:: ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt
Referenced by:: §18.24, §18.24.
Encodings:: BibTeX

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Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:: ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt
Referenced by:: §18.29.
Encodings:: BibTeX

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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R. Wong and Y. Zhao (2002b) Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.

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External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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19: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of

f(t)

is defined by … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For a Lebesgue–Stieltjes measure

\,\mathrm{d}\alpha

X

let

L^{2}\left(X,\,\mathrm{d}\alpha\right)

be the space of all Lebesgue–Stieltjes measurable complex-valued functions on

X

which are square integrable with respect to

\,\mathrm{d}\alpha

, ►

1.18.11

\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}\alpha(x)<\infty.

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.13

c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,

f\in L^{2}\left(X\right)

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Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral and $n$ : nonnegative integer
Referenced by:: §1.18(v), §1.18(vi), §1.18(ii)
Permalink:: http://dlmf.nist.gov/1.18.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.14

\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|c_{% n}\right|}^{2},

f\in L^{2}\left(X\right)

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Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

… ►Writing Hermite’s differential equation (see Tables 18.3.1 and 18.8.1) in the form above, the eigenfunctions are

{\mathrm{e}}^{-x^{2}/2}H_{n}\left(x\right)

(

H_{n}

a Hermite polynomial,

n=0,1,2,\ldots

), with eigenvalues

\lambda_{n}=2n+1\in\boldsymbol{\sigma}_{p}

, for the differential operator …