Lamé polynomials

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21: Bibliography S

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Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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ISSN 0020-7225, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(ii).

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22: Bibliography P

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A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:

Document, ZentralBlatt

Referenced by:

§29.18(iv).

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23: Errata

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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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24: 29.7 Asymptotic Expansions

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§29.7(i) Eigenvalues

… ►The same Poincaré expansion holds for $b_{\nu }^{m+1}\left(k^2\right)$, since … ►

§29.7(ii) Lamé Functions

… ►In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$ and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.

25: 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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26: Bibliography V

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

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Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.

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ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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27: 31.8 Solutions via Quadratures

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31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

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Annotations for §31.8 and Ch.31

… ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. … ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …When $\lambda =-4q$ approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. …

28: 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).

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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

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Document, ZentralBlatt

Referenced by:

§18.12.

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Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.

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Document, MathReview (F. L. Spitzer), ZentralBlatt

Referenced by:

§18.33(v).

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Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

3rd item.

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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(i).

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29: Bibliography I

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The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.15(ii), 1st item, §29.7(i).

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Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

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30: Bibliography W

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Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

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