DLMF: Untitled Document

§18.3 Definitions

… ►

3.

As given by a Rodrigues formula (18.5.5).

►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

,

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\dots,N+1

, of

T_{N+1}\left(x\right)

: … ►For another version of the discrete orthogonality property of the polynomials

T_{n}\left(x\right)

see (3.11.9). …

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

,

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations

E_{n}

,

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

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A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

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I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and

k

-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:: ISSN 0022-2488, Document, Link, MathReview Entry
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

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D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

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25.12.3

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{z}{z-1}% \right)=-\frac{1}{2}(\ln\left(1-z\right))^{2},

z\in\mathbb{C}\setminus[1,\infty)

.

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Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction, $x$ : real variable and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.4), p. 2808)
A&S Ref:: 27.7.5 (is a modified form with $z=1-x$ )
Permalink:: http://dlmf.nist.gov/25.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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25.12.4

\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{1}{z}% \right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2},

z\in\mathbb{C}\setminus[0,\infty)

.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction and $z$ : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.2), p. 2808)
Referenced by:: §25.12(ii)
Permalink:: http://dlmf.nist.gov/25.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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25.12.6

\operatorname{Li}_{2}\left(x\right)+\operatorname{Li}_{2}\left(1-x\right)=% \frac{1}{6}\pi^{2}-(\ln x)\ln\left(1-x\right),

0<x<1

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
Keywords:: connection formula
Source:: Maximon (2003, (3.3), p. 2808)
Permalink:: http://dlmf.nist.gov/25.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.12(i), §25.12 and Ch.25

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See accompanying text — Figure 25.12.1: Dilogarithm function $\operatorname{Li}_{2}\left(x\right)$ , $-20\leq x<1.$ Magnify

►

…

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V. Laĭ (1994) The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

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Notes:: In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)
External Links:: ISSN 0564-6162, Document, MathReview, ZentralBlatt
Referenced by:: §31.18.
Encodings:: BibTeX

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P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

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W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

…

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). …For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. … ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain

w(x)

. … ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. …

… ►

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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G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:: ISSN 0025-584X, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

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R. Wong and H. Y. Zhang (2009a) On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. (Singap.) 7 (4), pp. 419–448.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §32.11(v), §32.11(v), §32.13(ii).
Encodings:: BibTeX

►

R. Wong and H. Y. Zhang (2009b) On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

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External Links:: ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt
Referenced by:: §32.13(ii).
Encodings:: BibTeX

…

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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P. A. Clarkson and J. B. McLeod (1988) A connection formula for the second Painlevé transcendent. Arch. Rational Mech. Anal. 103 (2), pp. 97–138.

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External Links:: ISSN 0945-8396, Document, MathReview (Reinhard Schäfke), ZentralBlatt
Referenced by:: §32.11(ii).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

E. T. Copson (1933) An approximation connected with

e^{-x}

. Proc. Edinburgh Math. Soc. (2) 3, pp. 201–206.

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External Links:: Document, ZentralBlatt
Referenced by:: §8.11(v).
Encodings:: BibTeX

…

connection%20with%20orthogonal%20polynomials%20on%20the%20line

11—20 of 445 matching pages

11: 18.3 Definitions

§18.3 Definitions

12: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

13: Bibliography M

14: 25.12 Polylogarithms

15: Bibliography L

16: 18.40 Methods of Computation

§18.40(i) Computation of Polynomials

17: Bibliography W

18: 8 Incomplete Gamma and Related
Functions

19: 28 Mathieu Functions and Hill’s Equation

20: Bibliography C

connection%20with%20orthogonal%20polynomials%20on%20the%20line

11—20 of 445 matching pages

11: 18.3 Definitions

§18.3 Definitions

12: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

13: Bibliography M

14: 25.12 Polylogarithms

15: Bibliography L

16: 18.40 Methods of Computation

§18.40(i) Computation of Polynomials

17: Bibliography W

18: 8 Incomplete Gamma and Related Functions

19: 28 Mathieu Functions and Hill’s Equation

20: Bibliography C

18: 8 Incomplete Gamma and Related
Functions