Nevai–Blumenthal class

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1: 18.2 General Orthogonal Polynomials

… ►Nevai (1979, p.39) defined the class $𝒮$ of orthogonality measures with support inside $[-1,1]$ such that the absolutely continuous part $w(x)dx$ has $w$ in the Szegő class $𝒢$. … ►

The Nevai class $𝐌(a,b)$

►The class $𝐌(a,b)$ ($a>0$, $b\in ℝ$), introduced by Nevai (1979, p.10), consists of all orthogonality measures $d\mu$ such that the coefficients $\sqrt{{\beta }_n}$ and ${\alpha }_n$ in the recurrence relation (18.2.11_8) for the corresponding orthonormal OP’s satisfy …Therefore this class is also called the Nevai–Blumenthal class. …

2: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

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Orthogonal polynomials. Mem. Amer. Math. Soc. 18 (213), pp. v+185 pp..

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ISSN 0065-9266, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi), §18.2(xi).

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

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3: 18.32 OP’s with Respect to Freud Weights

… ►See the early survey by Nevai (1986, Part 2). …However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). …

4: 18.39 Applications in the Physical Sciences

… ►As the Coulomb–Pollaczek OP’s are members of the Nevai-Blumenthal class, this is, for , a physical example of the properties of the zeros of such OP’s, and their possible accumulation at $x=-1$, as discussed in §18.2(xi). …

5: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

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6: Bibliography B

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A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.

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Document, MathReview (S. Rosendorff), ZentralBlatt

Referenced by:

§15.18.

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Ueber die Entwickelung einer willkürlichen Function nach den Nennern des Kettenbruches für ${\int }_{-\mathrm{\infty }}^0\frac{\phi (\xi )d\xi }{z-\xi }$ .

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Notes:

Dissertation, Göttingen

Referenced by:

§18.2(xi).

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

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7: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

… ►These eight further families can be grouped in two classes of OP’s: ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

… ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). ►

Hahn, Krawtchouk, Meixner, and Charlier

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8: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

9: 22.22 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C13). …

10: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.