# relations to Lamé polynomials

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##### 1: 29.12 Definitions
###### §29.12(i) Elliptic-Function Form
There are eight types of Lamé polynomials, defined as follows: …
##### 2: 29.6 Fourier Series
In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … Consequently, $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
##### 3: 31.8 Solutions via Quadratures
the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … Here $\Psi_{g,N}(\lambda,z)$ 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. …(This $\nu$ is unrelated to the $\nu$ in §31.6.) … For $\mathbf{m}=(m_{0},0,0,0)$, 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. …
##### 4: 29.1 Special Notation
All derivatives are denoted by differentials, not by primes. The main functions treated in this chapter are the eigenvalues $a^{2m}_{\nu}\left(k^{2}\right)$, $a^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+2}_{\nu}\left(k^{2}\right)$, the Lamé functions $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$, and the Lamé polynomials $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$, $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$. The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … The relation to the Lamé functions $L^{(m)}_{c\nu}$, $L^{(m)}_{s\nu}$of Jansen (1977) is given by …The relation to the Lamé functions ${\rm Ec}^{m}_{\nu}$, ${\rm Es}^{m}_{\nu}$ of Ince (1940b) is given by …
##### 5: 29.14 Orthogonality
###### §29.14 Orthogonality
Lamé polynomials are orthogonal in two ways. First, the orthogonality relations (29.3.19) apply; see §29.12(i). …is orthogonal and complete with respect to the inner product … Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …
##### 6: Bibliography
• 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.
• R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
• ##### 7: Bibliography B
• S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.
• G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
• G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..
• T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.
• M. Brack, M. Mehta, and K. Tanaka (2001) Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.
• ##### 8: 28.34 Methods of Computation
• (b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

• (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 equation (§28.31(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: 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 $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ 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.
• ##### 10: Bibliography M
• N. W. Macfadyen and P. Winternitz (1971) Crossing symmetric expansions of physical scattering amplitudes: The $O(2,1)$ group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.
• C. Micu and E. Papp (2005) Applying $q$-Laguerre polynomials to the derivation of $q$-deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.
• M. Micu (1968) Recursion relations for the $3$-$j$ symbols. Nuclear Physics A 113 (1), pp. 215–220.
• G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
• S. C. Milne (1985c) A new symmetry related to $\mathit{SU}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.