multiple orthogonal polynomials

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11: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). … ►The polynomial …has $n$ zeros in $ℂ$, counting each zero according to its multiplicity. … ►

Example. Wilkinson’s Polynomial

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12: 3.2 Linear Algebra

… ►By repeatedly subtracting multiples of each row from the subsequent rows we obtain a matrix of the form … ►The polynomial …The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial (§3.8(i)). To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $𝐀$ is nondefective, that is, $𝐀$ has a complete set of $n$ linearly independent eigenvectors. …

13: Bibliography

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

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Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:

Link, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.3.

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

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Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.

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ISSN 0065-9266, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.28(iv).

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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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.

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14: Bibliography B

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(iii).

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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt

Referenced by:

§18.32.

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A uniform asymptotic formula for orthogonal polynomials associated with $\exp (-x^4)$ . J. Approx. Theory 98, pp. 146–166.

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ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.32.

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

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

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Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

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External Links:

ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§18.2(xii).

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15: Bibliography K

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Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.16(ii).

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Orthogonal polynomials and computer algebra. In Recent developments in complex analysis and computer algebra (Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.), Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

CAOP (website).

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Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:

MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.37(ii).

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Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

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Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

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ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§18.32.

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16: Bibliography C

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

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Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.

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ISSN 0305-4470, Document, MathReview (Tim H. Baker), ZentralBlatt

Referenced by:

§18.26(v), §18.29.

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An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 0-677-04150-0, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§1.16(ix), (18.2.33), (18.2.36), (18.2.38), §18.2(v), §18.2(iii), §18.2(iv), §18.2(vii), §18.2(x), §18.20(i), (18.30.25), §18.30(ii), §18.30, §18.40(ii), Ch.18.

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Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.28(iv).

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Basic hypergeometric functions and orthogonal Laurent polynomials. Proc. Amer. Math. Soc. 140 (6), pp. 2075–2089.

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ISSN 0002-9939, Link, MathReview (Roelof Koekoek), ZentralBlatt

Referenced by:

§18.33(iv).

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17: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

(18.33.17), (18.33.18), (18.33.21), (18.33.22), (18.33.23), (18.33.24), (18.33.25), (18.33.26), (18.33.27), (18.33.28), (18.33.30), (18.33.31), (18.33.32), §18.33(vi), §18.33(i), §18.33(vi).

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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§18.33(i), §18.33(vi).

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On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.

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External Links:

ISSN 0002-9939, Document, Link, MathReview (A. C. Offord)

Referenced by:

§18.35(i).

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Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.

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Notes:

American Mathematical Society Colloquium Publications, Vol. 23

External Links:

MathReview (J. Burlak)

Referenced by:

Ch.14, §9.1, Notations A.

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18: Bibliography M

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

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Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

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External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

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Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-82472-9, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.

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External Links:

ISSN 0377-0427, Document, MathReview (Leonid B. Golinskiĭ), ZentralBlatt

Referenced by:

§32.15.

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Exceptional orthogonal polynomials.

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Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

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19: 3.11 Approximation Techniques

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§3.11(i) Minimax Polynomial Approximations

… ►They enjoy an orthogonal property with respect to integrals: …When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ► … ►requires approximately $n^2$ multiplications. …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►If an eigenvalue has multiplicity $>1$, the eigenfunctions may always be orthogonalized in this degenerate sub-space. … ► … ►, of unit multiplicity, unless otherwise stated. … ►Letting $n$ run from $-\mathrm{\infty }$ to $\mathrm{\infty }$ this multiplicity change is automatically included: … ►Then orthogonality and normalization relations are …