characteristic polynomial

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1: 3.2 Linear Algebra

… ►is called the characteristic polynomial of $𝐀$ and its zeros are the eigenvalues of $𝐀$. The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial (§3.8(i)). … … ►Its characteristic polynomial can be obtained from the recursion ►

3.2.23 $$p_{k+1}(\lambda )=(\lambda -{\alpha }_{k+1})p_k(\lambda )-{\beta }_{k+1}^2{p}_{k-1}(\lambda ),$$ $k=0,1,\mathrm{\dots },n-1$,

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Defines:

$p_k(\lambda )$: characteristic polynomial (locally)

Symbols:

${\alpha }_j$: matrix, ${\beta }_j$: matrix and $\lambda$: eigenvalue

Referenced by:

§3.2(vi), §3.2(vi), Erratum (V1.1.3) for Subsection 3.2(vi)

Permalink:

http://dlmf.nist.gov/3.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §3.2(vi), §3.2 and Ch.3

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2: 28.34 Methods of Computation

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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3: 21.7 Riemann Surfaces

… ►where $P(\lambda ,\mu )$ is a polynomial in $\lambda$ and $\mu$ that does not factor over ${ℂ}^2$. … ►

§21.7(ii) Fay’s Trisecant Identity

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§21.7(iii) Frobenius’ Identity

… ►where $Q(\lambda )$ is a polynomial in $\lambda$ of odd degree $2g+1$ $(\ge 5)$. …

4: Bibliography L

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Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

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External Links:

ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

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The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

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External Links:

ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

BibTeX

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On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

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External Links:

FullText, MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

BibTeX

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

Encodings:

BibTeX

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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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External Links:

ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

Encodings:

BibTeX

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5: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

BibTeX

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A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

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Notes:

Reviewed in JFM 36.0512.01

External Links:

Document, ZentralBlatt

Referenced by:

§18.12.

Encodings:

BibTeX

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Mathieu’s Equation for Complex Parameters. Tables of Characteristic Values. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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External Links:

MathReview (L. Fox)

Referenced by:

3rd item.

Encodings:

BibTeX

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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6: Bibliography F

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

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External Links:

ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

Encodings:

BibTeX

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

BibTeX

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7: 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).

Encodings:

BibTeX

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Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:

ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§31.15(ii).

Encodings:

BibTeX

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Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

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External Links:

Document

Referenced by:

6th item.

Encodings:

BibTeX

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A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.

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External Links:

ISSN 0025-570X, MathReview Entry, ZentralBlatt

Referenced by:

§24.5(ii), Ch.24.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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8: Bibliography C

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A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

BibTeX

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Stability properties of disk polynomials. Numer. Algorithms.

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External Links:

Document

Referenced by:

(18.14.3_5).

Encodings:

BibTeX

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Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)

Referenced by:

§18.32.

Encodings:

BibTeX

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

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.

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Notes:

Includes double-precision Fortran program, maximum accuracy 13D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

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9: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). …