algebraic Lamé functions

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11: 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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12: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

13: 28.34 Methods of Computation

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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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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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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§28.34(iv) Modified Mathieu Functions

►For the modified functions we have: …

14: 29.15 Fourier Series and Chebyshev Series

§29.15 Fourier Series and Chebyshev Series

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Polynomial ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$

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Polynomial ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$

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§29.15(ii) Chebyshev Series

… ►For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).

15: Bibliography M

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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).

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Crossing symmetric expansions of physical scattering amplitudes: The $O(2,1)$ group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.

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External Links:

Document, MathReview

Referenced by:

§29.19(ii).

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

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Magma

Referenced by:

4th item.

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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt

Referenced by:

§32.9(iii).

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Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).

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Notes:

English translation: Lamé wave functions of the ellipsoid of revolution appeared as NACA Technical Memorandum No. 1224, 1949

External Links:

MathReview

Referenced by:

§30.9(iii).

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16: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►The Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, $m=0,1,\mathrm{\dots },\nu$, and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, $m=1,2,\mathrm{\dots },\nu$, are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of $z$. … ►

§29.12(ii) Algebraic Form

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17: Bibliography

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Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.

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External Links:

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

Referenced by:

§29.15(ii), 2nd item.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

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Linear algebra done right. Third edition, Undergraduate Texts in Mathematics, Springer, Cham.

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External Links:

ISBN 978-3-319-11079-0; 978-3-319-11080-6, Document, Link, MathReview Entry

Referenced by:

§1.2(v), §1.2(vi), §1.3(iv).

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18: Bibliography I

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Theta Functions. Springer-Verlag, New York.

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Notes:

Die Grundlehren der mathematischen Wissenschaften, Band 194

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ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt

Referenced by:

§21.5(ii), §21.8, Ch.21.

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

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Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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ZentralBlatt

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4th item, 2nd item.

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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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19: Bibliography D

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Tables of Higher Mathematical Functions I. Principia Press, Bloomington, Indiana.

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations P.

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

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3rd item.

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Abstract Algebra. 2nd edition, Prentice Hall Inc., Englewood Cliffs, N.J..

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ISBN 0-13-569302-0, MathReview (N. Popescu), ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iii).

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Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

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Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

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ISSN 1029-8479, Document, MathReview, ZentralBlatt

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§22.11.

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Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.

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ISSN 0035-7596, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§18.34(ii).

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20: Bibliography B

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Algebraic Combinatorics. I: Association Schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA.

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External Links:

ISBN 0-8053-0490-8, MathReview Entry, ZentralBlatt

Referenced by:

§18.28(xi).

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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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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

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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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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(i).

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Plane Algebraic Curves. Birkhäuser Verlag, Basel.

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Notes:

Translated from the German by John Stillwell

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ISBN 3-7643-1769-8, MathReview, ZentralBlatt

Referenced by:

§21.7(i).

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