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Published online by Cambridge University Press: 15 September 2014
In a previous paper the author has employed certain formulæ of Dr Dougall's connecting the Associated Legendre Functions Pnm, where m is a positive integer and n is not integral, with the polynomials Ppm in which p is a positive integer, to deduce the Addition Theorem for the Legendre Functions of the first kind from the corresponding theorem for the Legendre Polynomials.
page 30 note * Proc. Edin. Math. Soc., vol. xlii (1924), pp. 93, 94Google Scholar.
page 30 note † Ibid., vol. xviii (1900), p. 78.
page 30 note ‡ The Associated Legendre Functions employed in this paper are those defined by Hobson, Phil. Trans., vol. clxxxvii, A (1896), pp. 443–531Google Scholar. These definitions, and many of the formulæ employed in this paper, are given in the author's Functions of a Complex Variable, chapter xv.
page 30 note § This form of the addition theorem can be deduced from the formula
where
as given in Whittaker and Watson's Analysis, p. 322, by writing ω = ϕ + π and using the formula Γ(n + m+1)Pn -m(z) = Γ(n-m +l)Pnm(z), which holds when m is an integer.
page 30 note ‖ Leipzig, Abh., 1886, p. 401.
page 31 note * Proc. Edin. Math. Soc., vol. xli (1923), pp. 82–90Google Scholar ; vol. xlii (1924), pp. 84-87. The symbol ∼ in the formula f(ζ)∼kϕ(ζ) is employed to denote that Lim f(ζ),/ϕ(ζ)=k. (See Bromwich's Infinite Series, Second Edition, p. 4.)
page 34 note * Cf. Proc. Edin. Math. Soc., vol. xli (1923), pp. 90, 91Google Scholar, and the author's Functions of a Complex Variable, p. 265.