relation to Lamé functions
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1: 31.7 Relations to Other Functions
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§31.7(ii) Relations to Lamé Functions
…2: 29.12 Definitions
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§29.12(i) Elliptic-Function Form
… βΊThere are eight types of Lamé polynomials, defined as follows: …3: 31.8 Solutions via Quadratures
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βΊFor , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form.
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4: 29.6 Fourier Series
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βΊIn addition, if satisfies (29.6.2), then (29.6.3) applies.
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βΊConsequently, reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).
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5: 29.1 Special Notation
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βΊThe relation to the Lamé functions
, of Jansen (1977) is given by
…The relation to the Lamé functions
, of Ince (1940b) is given by
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6: 29.2 Differential Equations
7: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
…8: Bibliography
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Some series of the zeta and related functions.
Analysis (Munich) 18 (2), pp. 131–144.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Tables of Lamé Polynomials.
Pergamon Press, The Macmillan Co., New York.
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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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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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9: Errata
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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions
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Source citations
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Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function
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Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)
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Chapter 25 Zeta and Related Functions
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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, and the Laguerre polynomial, , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23
Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.
The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.
A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.