multiple orthogonal polynomials

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1: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►

§18.36(iii) Multiple Orthogonal Polynomials

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2: 18.18 Sums

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§18.18(iii) Multiplication Theorems

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3: 31.9 Orthogonality

§31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

… ►For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). …

4: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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5: 18.27 $q$-Hahn Class

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§18.27(i) Introduction

… ►All these systems of OP’s have orthogonality properties of the form … ►

§18.27(ii) $q$-Hahn Polynomials

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§18.27(iii) Big $q$-Jacobi Polynomials

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Limit Relations

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6: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

7: Bibliography D

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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:

ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§18.38(ii), §18.38(ii).

Encodings:

BibTeX

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Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (12), pp. 1491–1552.

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External Links:

ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.

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External Links:

ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.12(iv).

Encodings:

BibTeX

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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.

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External Links:

ISSN 0036-1410, Document, MathReview (Kathi Karima Selig), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

BibTeX

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8: 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 ${\mathrm{𝐻𝑝}}_{n,m}\left(z\right)$. ►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 $ℰ$. … ►

§31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …

9: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

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Orthogonality

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§18.28(viii) $q$-Racah Polynomials

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10: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … ►

The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

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