generalized Bessel polynomials

(0.008 seconds)

21—30 of 52 matching pages

21: Bibliography T

… ►

A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

ⓘ

External Links:: ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt
Referenced by:: §35.4(i), §35.9.
Encodings:: BibTeX

►

J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

ⓘ

Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

N. M. Temme (1986) Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

ⓘ

External Links:: FullText
Referenced by:: §2.4(vi).
Encodings:: BibTeX

… ►

N. M. Temme (1990a) Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §18.11(i), §18.16(vi).
Encodings:: BibTeX

… ►

N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

ⓘ

External Links:: ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt
Referenced by:: §24.11, §24.16(i).
Encodings:: BibTeX

…

22: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

►

§10.41(i) Asymptotic Forms

… ► … ►

§10.41(iv) Double Asymptotic Properties

… ►

23: Bibliography G

… ►

B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

ⓘ

External Links:: ISSN 0771-050X, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

W. Gautschi (1993) On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

ⓘ

External Links:: ISSN 0010-4655, Document, MathReview, ZentralBlatt
Referenced by:: §25.12(iii), §25.18(i), 5th item.
Encodings:: BibTeX

… ►

V. X. Genest, L. Vinet, and A. Zhedanov (2016) The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

ⓘ

External Links:: ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt
Referenced by:: §18.28(xi).
Encodings:: BibTeX

… ►

M. L. Glasser (1979) A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview
Referenced by:: §10.71(ii).
Encodings:: BibTeX

… ►

E. Grosswald (1978) Bessel Polynomials. Lecture Notes in Mathematics, Vol. 698, Springer, Berlin-New York.

ⓘ

External Links:: ISBN 3-540-09104-1, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §10.49(ii), (18.34.1), (18.34.2), (18.34.6).
Encodings:: BibTeX

…

24: 14.15 Uniform Asymptotic Approximations

… ►In other words, the convergent hypergeometric series expansions of

\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)

are also generalized (and uniform) asymptotic expansions as

\mu\to\infty

, with scale

\ifrac{1}{\Gamma\left(j+1+\mu\right)}

j=0,1,2,\dots

; compare §2.1(v). … ►Here

I

and

K

are the modified Bessel functions (§10.25(ii)). … ►

14.15.11

\mathsf{P}^{-\mu}_{\nu}\left(\cos\theta\right)=\frac{1}{\nu^{\mu}}\left(\frac{% \theta}{\sin\theta}\right)^{1/2}\left(J_{\mu}\left(\left(\nu+\tfrac{1}{2}% \right)\theta\right)+O\left(\frac{1}{\nu}\right)\operatorname{env}\mskip-2.0% muJ_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)\right),

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\operatorname{env}\mskip-2.0muJ_{\NVar{\nu}}\left(\NVar{x}\right)$ : envelope of Bessel function $J_{\NVar{\nu}}\left(\NVar{x}\right)$ , $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/14.15.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(iii), §14.15 and Ch.14

… ►For the Bessel functions

J

and

Y

see §10.2(ii), and for the

\operatorname{env}

functions associated with

J

and

Y

see §2.8(iv). … ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials

P_{n}\left(\cos\theta\right)

n\to\infty

with

\theta

fixed. …

25: 18.39 Applications in the Physical Sciences

… ►The functions

\phi_{n}

are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). The finite system of functions

\psi_{n}

is orthonormal in

L^{2}\left(\mathbb{R},\,\mathrm{d}x\right)

, see (18.34.7_3). … ►The associated Coulomb–Laguerre polynomials are defined as … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

… ►

The Coulomb–Pollaczek Polynomials

…

26: Bibliography B

… ►

W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

►

W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

ⓘ

External Links:: MathReview
Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
Encodings:: BibTeX

… ►

P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

ⓘ

External Links:: ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §9.13(ii).
Encodings:: BibTeX

… ►

C. Brezinski (1980) Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

ⓘ

External Links:: ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt
Referenced by:: §3.11(iv).
Encodings:: BibTeX

…

27: 18.38 Mathematical Applications

… ►

Approximation Theory

… ►

Integrable Systems

… ►The Askey–Gasper inequality … ►If we consider this abstract algebra with additional relation (18.38.9) and with dependence on

a, b, c, d

according to (18.38.7) then it is isomorphic with the algebra generated by

K_{0}=L

given by (18.28.6_2),

(K_{1}f)(z)=(z+z^{-1})f(z)

and

K_{2}

given by (18.38.4), and

K_{0},K_{1},K_{2}

act on the linear span of the Askey–Wilson polynomials (18.28.1). … ► …

28: Bibliography K

… ►

R. P. Kanwal (1983) Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

ⓘ

Notes:: Theory and technique
External Links:: ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt
Referenced by:: §1.16(viii).
Encodings:: BibTeX

… ►

T. Kasuga and R. Sakai (2003) Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

ⓘ

External Links:: ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

K. S. Kölbig (1986) Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §25.12(ii).
Encodings:: BibTeX

… ►

T. H. Koornwinder and F. Bouzeffour (2011) Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

ⓘ

External Links:: ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt
Referenced by:: §18.38(iii), §18.38(iii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2012) Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

ⓘ

Notes:: Electronic only
External Links:: Document, FullText
Referenced by:: §18.28(i), §18.28(i).
Encodings:: BibTeX

…

29: Bibliography I

… ►

Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(iii).
Encodings:: BibTeX

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

ⓘ

External Links:: ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt
Referenced by:: §18.39(v).
Encodings:: BibTeX

… ►

M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

…

30: Bibliography E

… ►

Á. Elbert (2001) Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

E. Elizalde (1986) An asymptotic expansion for the first derivative of the generalized Riemann zeta function. Math. Comp. 47 (175), pp. 347–350.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (S. Audinarayana Moorthy), ZentralBlatt
Referenced by:: (25.11.44), (25.11.45), §25.11(xii), §25.11(xii).
Encodings:: BibTeX

… ►

D. Elliott (1971) Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §18.15(i).
Encodings:: BibTeX

… ►

R. Ernvall (1979) Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku. Ser. A I 178, pp. 1–72.

ⓘ

External Links:: ISSN 0082-7002, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: §24.10(ii).
Encodings:: BibTeX

… ►

W. D. Evans, W. N. Everitt, K. H. Kwon, and L. L. Littlejohn (1993) Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §18.34(ii).
Encodings:: BibTeX

…