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The sign of Lommel's function


Author: J. Steinig
Journal: Trans. Amer. Math. Soc. 163 (1972), 123-129
MSC: Primary 33.25
DOI: https://doi.org/10.1090/S0002-9947-1972-0284625-X
MathSciNet review: 0284625
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Abstract: Lommel's function $ {s_{\mu ,\nu }}(x)$ is a particular solution of the differential equation $ {x^2}y'' + xy' + ({x^2} - {\nu ^2})y = {x^{\mu + 1}}$. It is shown here that $ {s_{\mu ,\nu }}(x) > 0$ for $ x > 0$, if $ \mu = \tfrac{1}{2}$ and $ \vert\nu \vert < \tfrac{1}{2}$, or if $ \mu > \tfrac{1}{2}$ and $ \vert\nu \vert \leqq \mu $. This includes earlier results of R. G. Cooke's. The sign of $ {s_{\mu ,\nu }}(x)$ for other values of $ \mu $ and $ \nu $ is also discussed.


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  • [1] A. W. Babister, Transcendental functions satisfying nonhomogeneous linear differential equations, Macmillan, New York, 1967. MR 34 #6158. MR 0206339 (34:6158)
  • [2] R. G. Cooke, Gibbs's phenomenon in Fourier-Bessel series and integrals, Proc. London Math. Soc. (2) 27 (1928), 171-192.
  • [3] -, A monotonic property of Bessel functions, J. London Math. Soc. 12 (1937), 180-185.
  • [4] -, On the sign of Lommel's function, J. London Math. Soc. 7 (1932), 281-283.
  • [5] W. Leighton, Ordinary differential equations, 3rd ed., Wadsworth, Belmont, Calif., 1970. MR 0259205 (41:3847)
  • [6] E. Lommel, Ueber eine mit den Besseischen Functionen verwandte Function, Math. Ann. 9 (1876), 425-444.
  • [7] E. Makai, On a monotonic property of certain Sturm-Liouville functions, Acta Math. Acad. Sci. Hungar. 3 (1952), 165-172. MR 14, 872. MR 0054103 (14:872e)
  • [8] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR 21 #5029. MR 0106295 (21:5029)
  • [9] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge; Macmillan, New York, 1944. MR 6, 64. MR 0010746 (6:64a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0284625-X
Keywords: Lommel functions, Bessel functions, changes of sign, oscillation theorems, inhomogeneous Bessel equation
Article copyright: © Copyright 1972 American Mathematical Society

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