# characteristic polynomial

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##### 1: 3.2 Linear Algebra
is called the characteristic polynomial of $\mathbf{A}$ and its zeros are the eigenvalues of $\mathbf{A}$. The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial3.8(i)). … … Its characteristic polynomial can be obtained from the recursion
3.2.23 $p_{k+1}(\lambda)=(\lambda-\alpha_{k+1})p_{k}(\lambda)-\beta_{k+1}^{2}p_{k-1}(% \lambda),$ $k=0,1,\dots,n-1$,
##### 2: 28.34 Methods of Computation
• (f)

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)).

• ##### 3: 21.7 Riemann Surfaces
where $P(\lambda,\mu)$ is a polynomial in $\lambda$ and $\mu$ that does not factor over ${\mathbb{C}}^{2}$. …
###### §21.7(iii) Frobenius’ Identity
where $Q(\lambda)$ is a polynomial in $\lambda$ of odd degree $2g+1$ $(\geq 5)$. …
##### 4: Bibliography L
• W. R. Leeb (1979) Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.
• D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.
• D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.
• J. L. López and N. M. Temme (1999a) Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.
• J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
• ##### 5: Bibliography B
• W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.
• H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.
• G. Blanch and D. S. Clemm (1969) Mathieu’s Equation for Complex Parameters. Tables of Characteristic Values. U.S. Government Printing Office, Washington, D.C..
• G. Blanch and I. Rhodes (1955) Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.
• W. Bühring (1994) The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.
• ##### 6: Bibliography F
• J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.
• J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.
• J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.
• N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.
• Y. Fukui and T. Horiguchi (1992) 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.
• ##### 7: Bibliography
• W. A. Al-Salam and L. Carlitz (1965) Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.
• M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
• F. Alhargan and S. Judah (1992) Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.
• T. M. Apostol (2008) A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.
• R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
• ##### 8: Bibliography C
• L. Carlitz (1954b) A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.
• J. M. Carnicer, E. Mainar, and J. M. Peña (2020) Stability properties of disk polynomials. Numer. Algorithms.
• P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
• P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.
• D. S. Clemm (1969) Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.
• ##### 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)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)$18.3). …