Remarks on the Zeros of the Associated Legendre Functions with Integral Degree
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Remarks on the Zeros of the Associated Legendre Functions with Integral Degree J.F. van Diejen
Published online: 3 October 2007 © Springer Science+Business Media B.V. 2007
Abstract We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order. Keywords Hypergeometric functions · Integrable systems · Inverse scattering theory · Solitons 1 Introduction The associated Legendre functions (or spherical functions) are given explicitly by [1, 7] 1 + tanh(x) z/2 1 − tanh(x) (1 − z)Pnz (tanh(x)) = F −n, n + 1; 1 − z; , 1 − tanh(x) 2 =
exp(zx) F (−n, −n − z; 1 − z; −e−2x ), (1 + e−2x )n
(1.1)
where (−) refers to the Euler gamma function and F (−, −; −; −) to the Gauss hypergeometric series, and where we have fixed the normalization such that no gamma factors appear in front of the hypergeometric series representation. When the degree n is integral the hypergeometric series on the first line of (1.1) terminates, whence it is then polynomial in the argument tanh(x) and—with the present normalization—rational in the order z (with poles at z = j , j = 1, . . . , n). The purpose of the present note is to characterize the locations of zeros of the integral-degree associated Legendre functions with respect to the order z. In [3, 4] detailed information concerning the locations of the zeros of the associated Legendre functions with respect to the argument was provided for the situation that the order and the degree differ by a positive integer. PreviWork supported in part by the ‘Anillo Ecuaciones Asociadas a Reticulados’, financed by the World Bank through the ‘Programa Bicentenario de Ciencia y Tecnología’, and by the ‘Programa Reticulados y Ecuaciones’ of the Universidad de Talca. J.F. van Diejen () Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile e-mail: [email protected]
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ously, a numerical study of the locations of the zeros with respect to the degree (for integral order and various values of the argument) was presented in [2]. Questions about the zeros of the associated Legendre functions fit within a rich tradition of research concerning the locations of the zeros of orthogonal polynomials [8, 9]. An important difference is, however, that in the present context we are not in the position to exploit an orthogonality structure. Instead, the idea behind the methods below is to employ a connection with the (inverse) scattering theory of the one-dimensional Schrödinger equation with Pöschl-Teller potential [6], which reveals that the integral-degree associated Legendre functions correspond to reflectionless wave functions that can be expressed in terms of the tau functions of the Korteweg-de Vries hierarchy [11]. Detailed information on the zeros can then be obtained from [10, 12], where the behavior of the zeros of such reflectionless wave functions was studied with the aid of (Ruijsenaars-Schneider type) integrable particle systems (cf. also [5], where information on the zeros
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