# relations to Lamé functions

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##### 2: 29.12 Definitions
###### §29.12(i) Elliptic-Function Form
There are eight types of Lamé polynomials, defined as follows: …
##### 3: 31.8 Solutions via Quadratures
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. …
##### 4: 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). …
##### 5: 29.1 Special Notation
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 …
##### 6: 29.2 Differential Equations
For the Weierstrass function $\wp$ see §23.2(ii). …
##### 7: Bibliography
• V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.
• 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.
• ##### 8: 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.
• ##### 9: 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). Secondly, the system of functions …is orthogonal and complete with respect to the inner product …
##### 10: Bibliography S
• R. Shail (1978) Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
• R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.
• 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 (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.
• B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.