associated Jacobi polynomials

♦ 11—20 of 23 matching pages ♦

(0.004 seconds)

11—20 of 23 matching pages

11: Bibliography

… ►

R. Askey and J. Wimp (1984) Associated Laguerre and Hermite polynomials. Proc. Roy. Soc. Edinburgh 96A, pp. 15–37.

ⓘ

External Links:: MathReview Entry
Referenced by:: (18.30.16), (18.30.17), (18.30.18), (18.30.19), (18.30.20), §18.30(iii), §18.30(iv), §18.30, §18.30(v), (18.35.10).
Encodings:: BibTeX

… ►

R. Askey and J. Fitch (1969) Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

ⓘ

External Links:: Document, MathReview (D. L. Colton), ZentralBlatt
Referenced by:: §18.17(iv).
Encodings:: BibTeX

►

R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

ⓘ

External Links:: FullText, MathReview (F. Schipp), ZentralBlatt
Referenced by:: §16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.
Encodings:: BibTeX

… ►

R. Askey and B. Razban (1972) An integral for Jacobi polynomials. Simon Stevin 46, pp. 165–169.

ⓘ

External Links:: MathReview (H. M. Srivastava), ZentralBlatt
Referenced by:: §18.17(viii).
Encodings:: BibTeX

►

R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

ⓘ

External Links:: ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.
Encodings:: BibTeX

…

12: 18.2 General Orthogonal Polynomials

… ► … ►

§18.2(vi) Zeros

… ►

§18.2(x) Orthogonal Polynomials and Continued Fractions

… ►Define the first associated monic orthogonal polynomials

p_{n}^{(1)}(x)

as monic OP’s satisfying … ► …

13: 18.1 Notation

… ►

Classical OP’s

►

Jacobi: $P^{(\alpha,\beta)}_{n}\left(x\right)$ .

… ►

Big $q$ -Jacobi: $P_{n}\left(x;a,b,c;q\right)$ .

►

Little $q$ -Jacobi: $p_{n}\left(x;a,b;q\right)$ .

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

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. Wilson), introduced the Askey-Wilson polynomials. …Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. … ►Askey was a member of the original editorial committee for the DLMF project, serving as an Associate Editor advising on all aspects of the project from the mid-1990’s to the mid-2010’s when the organizational structure of the DLMF project was reconstituted; see About the Project.

15: Bibliography C

… ►

Y. Chow, L. Gatteschi, and R. Wong (1994) A Bernstein-type inequality for the Jacobi polynomial. Proc. Amer. Math. Soc. 121 (3), pp. 703–709.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.14(i).
Encodings:: BibTeX

… ►

P. A. Clarkson and E. L. Mansfield (2003) The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.

ⓘ

External Links:: ISSN 0951-7715, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

… ►

P. A. Clarkson (2003a) The third Painlevé equation and associated special polynomials. J. Phys. A 36 (36), pp. 9507–9532.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §18.38(iii), §32.8(iii), §32.9(i).
Encodings:: BibTeX

►

P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

►

P. A. Clarkson (2005) Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Davide Batic), ZentralBlatt
Referenced by:: §32.8(v), §32.9(ii).
Encodings:: BibTeX

…

16: 14.28 Sums

§14.28 Sums

►

§14.28(i) Addition Theorem

… ►

14.28.1

P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)% ^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum% _{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1% \right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cos\NVar{z}$ : cosine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, $z$ : complex variable, $m$ : nonnegative integer and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.28.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.28(i), §14.28 and Ch.14

… ►

§14.28(ii) Heine’s Formula

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. …

17: Bibliography I

… ►

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, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

ⓘ

External Links:: ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.26(iv).
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 (1986) Asymptotics of the Askey-Wilson and

q

-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §18.29.
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

…

18: 31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►The case

\alpha=-n

for nonnegative integer

n

corresponds to the Heun polynomial

\mathit{Hp}_{n,m}\left(z\right)

. ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse

\mathcal{E}

. … ►

\mu=\gamma+\delta-2.

►In each case

P_{j}^{6}

can be expressed in terms of a Jacobi polynomial (§18.3). …

19: Bibliography D

… ►

G. Delic (1979a) Chebyshev expansion of the associated Legendre polynomial

P^{M}_{L}(x)

. Comput. Phys. Comm. 18 (1), pp. 63–71.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 25D
External Links:: Document, Code
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

C. F. Dunkl (1989) Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.

ⓘ

External Links:: ISSN 0002-9947, Link, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

T. M. Dunster (1999) Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §15.12(iii), §18.15(i), §18.15(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (2003b) Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

ⓘ

External Links:: ISSN 0308-2105, Document, MathReview, ZentralBlatt
Referenced by:: §14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.
Encodings:: BibTeX

►

T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

ⓘ

External Links:: ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §14.15(iii), §2.8(i).
Encodings:: BibTeX

…

20: 15.9 Relations to Other Functions

… ►

Jacobi

… ►

§15.9(ii) Jacobi Function

►This is a generalization of Jacobi polynomials (§18.3) and has the representation … ►The Jacobi transform is defined as … …