second order differential operators

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11: 18.27 $q$ -Hahn Class

… ►The

q

-Hahn class OP’s comprise systems of OP’s

\{p_{n}(x)\}

n=0,1,\dots,N

, or

n=0,1,2,\dots

, that are eigenfunctions of a second order

q

-difference operator. …In the

q

-Hahn class OP’s the role of the operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the

q

-derivative

\mathcal{D}_{q}

, as defined in (17.2.41). … ►

18.27.16

\int_{0}^{\infty}L^{(\alpha)}_{n}\left(x;q\right)L^{(\alpha)}_{m}\left(x;q% \right)\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}\,\mathrm{d}x=\frac{\left(% q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(1)}\delta_{n,m},

\alpha>-1

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : $q$ -Laguerre polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(v), §18.27 and Ch.18

… ►

18.27.19

\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\,\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : Stieltjes–Wigert polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.27.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vi), §18.27 and Ch.18

… ►

18.27.20

\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)\,% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : Stieltjes–Wigert polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.27.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vi), §18.27 and Ch.18

…

12: Bibliography K

… ►

T. H. Koornwinder (2006) Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

ⓘ

External Links:: FullText, MathReview Entry, ZentralBlatt
Referenced by:: §18.9(iii).
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

►

T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

ⓘ

External Links:: ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)
Referenced by:: §18.14(i).
Encodings:: BibTeX

… ►

J. J. Kovacic (1986) An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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External Links:: ISSN 0747-7171, MathReview, ZentralBlatt
Referenced by:: §31.14(ii).
Encodings:: BibTeX

… ►

K. H. Kwon, L. L. Littlejohn, and G. J. Yoon (2006) Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §18.36(i).
Encodings:: BibTeX

13: 10.40 Asymptotic Expansions for Large Argument

… ►

$\nu$ -Derivative

… ►

§10.40(ii) Error Bounds for Real Argument and Order

… ►

§10.40(iii) Error Bounds for Complex Argument and Order

… ►where

\mathcal{V}

denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that

|\Re t|

changes monotonically. Bounds for

\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)

are given by …

14: 16.8 Differential Equations

§16.8 Differential Equations

►

§16.8(i) Classification of Singularities

… ► … ►In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected. … ►For other values of the

b_{j}

, series solutions in powers of

z

(possibly involving also

\ln z

) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. …

15: 18.36 Miscellaneous Polynomials

… ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the

L_{n}^{(-k)}(x)

polynomials, self-adjointness implying both orthogonality and completeness. … ►Exceptional type I

X_{m}

-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order

m

, or, said another way, the first

m

polynomial orders,

0,1,\ldots,m-1

are missing. … ►The

y(x)=\hat{L}^{(k)}_{n}\left(x\right)

satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: … ►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).

16: 1.17 Integral and Series Representations of the Dirac Delta

… ►for all functions

\phi(x)

that are continuous when

x\in(-\infty,\infty)

, and for each

a

\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x

converges absolutely for all sufficiently large values of

n

. … ►More generally, assume

\phi(x)

is piecewise continuous (§1.4(ii)) when

x\in[-c,c]

for any finite positive real value of

c

, and for each

a

\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x

converges absolutely for all sufficiently large values of

n

. … ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►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)): …

17: Bibliography S

… ►

R. B. Shirts (1993a) The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. ACM Trans. Math. Software 19 (3), pp. 377–390.

ⓘ

External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: item (e).
Encodings:: BibTeX

… ►

R. Spigler and M. Vianello (1992) Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

►

R. Spigler and M. Vianello (1997) A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

… ►

R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

ⓘ

External Links:: ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

►

R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

ⓘ

External Links:: ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

…

18: Bibliography R

… ►

M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

ⓘ

External Links:: ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

S. R. Rengarajan and J. E. Lewis (1980) Mathieu functions of integral orders and real arguments. IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.

ⓘ

Notes:: Includes Fortran program
External Links:: FullText
Referenced by:: §28.36(ii), §28.36(iii).
Encodings:: BibTeX

… ►

S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

ⓘ

External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

… ►

G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

ⓘ

External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

… ►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

ⓘ

External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

19: Bibliography J

… ►

M. Jimbo and T. Miwa (1981) Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

ⓘ

External Links:: ISSN 0167-2789, Document, MathReview (V. A. Golubeva), ZentralBlatt
Referenced by:: §32.4(i), §32.4(ii), §32.4(iv), §32.4(v), §32.6(ii), §32.6(iii), §32.6(iv), §32.6(v), §32.6(v), §32.6(v).
Encodings:: BibTeX

… ►

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

ⓘ

Notes:: Second edition of Jones and Sleeman (2003)
External Links:: ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt
Referenced by:: Profile Brian D. Sleeman.
Encodings:: BibTeX

►

D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

ⓘ

Notes:: A second edition was published in 2010.
External Links:: ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt
Referenced by:: D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).
Encodings:: BibTeX

… ►

D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.10(vi).
Encodings:: BibTeX

… ►

B. R. Judd (1998) Operator Techniques in Atomic Spectroscopy. Princeton University Press, Princeton, NJ.

ⓘ

External Links:: ISBN 0-691-05901-2
Referenced by:: §34.12, Profile Brian R. Judd.
Encodings:: BibTeX

…

20: 3.4 Differentiation

… ►

First-Order

… ►

Second-Order

… ►

Fourth-Order

… ►

Biharmonic Operator

… ►Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …