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1: 29.2 Differential Equations
For the Weierstrass function see §23.2(ii). …
2: 31.8 Solutions via Quadratures
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. …
3: 29.12 Definitions
§29.12 Definitions
§29.12(i) Elliptic-Function Form
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
4: Bibliography
  • H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • F. M. Arscott and I. M. Khabaza (1962) Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • F. M. Arscott (1964b) 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.
  • 5: 31.7 Relations to Other Functions
    §31.7 Relations to Other Functions
    §31.7(i) Reductions to the Gauss Hypergeometric Function
    §31.7(ii) Relations to Lamé Functions
    equation (31.2.1) becomes Lamé’s equation with independent variable ζ ; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …
    6: 29.6 Fourier Series
    §29.6 Fourier Series
    §29.6(i) Function Ec ν 2 m ( z , k 2 )
    In addition, if H satisfies (29.6.2), then (29.6.3) applies. … Consequently, Ec ν 2 m ( z , k 2 ) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
    §29.6(ii) Function Ec ν 2 m + 1 ( z , k 2 )
    7: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
  • S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
  • B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.
  • B. I. Suleĭmanov (1987) The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).
  • 8: 28.34 Methods of Computation
    §28.34(i) Characteristic Exponents
  • (c)

    Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

  • (d)

    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).

  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • (d)

    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).

  • 9: 29.1 Special Notation
    All derivatives are denoted by differentials, not by primes. The main functions treated in this chapter are the eigenvalues a ν 2 m ( k 2 ) , a ν 2 m + 1 ( k 2 ) , b ν 2 m + 1 ( k 2 ) , b ν 2 m + 2 ( k 2 ) , the Lamé functions Ec ν 2 m ( z , k 2 ) , Ec ν 2 m + 1 ( z , k 2 ) , Es ν 2 m + 1 ( z , k 2 ) , Es ν 2 m + 2 ( z , k 2 ) , and the Lamé polynomials uE 2 n m ( z , k 2 ) , sE 2 n + 1 m ( z , k 2 ) , cE 2 n + 1 m ( z , k 2 ) , dE 2 n + 1 m ( z , k 2 ) , scE 2 n + 2 m ( z , k 2 ) , sdE 2 n + 2 m ( z , k 2 ) , cdE 2 n + 2 m ( z , k 2 ) , scdE 2 n + 3 m ( z , k 2 ) . … Other notations that have been used are as follows: Ince (1940a) interchanges a ν 2 m + 1 ( k 2 ) with b ν 2 m + 1 ( k 2 ) . The relation to the Lamé functions L c ν ( m ) , L s ν ( m ) of Jansen (1977) is given by …The relation to the Lamé functions Ec ν m , Es ν m of Ince (1940b) is given by …
    10: Bibliography W
  • X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • B. M. Watrasiewicz (1967) Some useful integrals of Si ( x ) , Ci ( x ) and related integrals. Optica Acta 14 (3), pp. 317–322.
  • J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.
  • J. Wimp (1984) Computation with Recurrence Relations. Pitman, Boston, MA.