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Monotonic sequences related to zeros of Bessel functions. (English) Zbl 1171.33304
Some inequalities concerning the values of the Bessel functions are proved by A. Akiyama and Y. Tanigawa in the course of their work on Salem numbers and uniform distribution modulo 1. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Such results by differential equations methods are derived in this paper.
MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
References:
[1]Akiyama, S., Tanigawa, Y.: Salem numbers and uniform distribution modulo 1. Publ. Math. (Debr.) 64, 329–341 (2004)
[2]Elbert, Á.: An approximation for the zeros of Bessel functions. Numer. Math. 59, 647–657 (1991) · Zbl 0760.65020 · doi:10.1007/BF01385801
[3]Elbert, Á.: Some recent results on the zeros of Bessel functions and orthogonal polynomials. Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999). J. Comput. Appl. Math. 133(1–2), 65–83 (2001) · Zbl 0989.33004 · doi:10.1016/S0377-0427(00)00635-X
[4]Elbert, Á., Gatteschi, L., Laforgia, A.: On the concavity of zeros of Bessel functions. Appl. Anal. 16, 261–278 (1983) · Zbl 0512.33003 · doi:10.1080/00036818308839474
[5]Elbert, Á., Laforgia, A.: On the square of the zeros of Bessel functions. SIAM J. Math. Anal. 15, 206–212 (1984) · Zbl 0541.33001 · doi:10.1137/0515017
[6]Elbert, Á., Laforgia, A.: Monotonicity properties of the zeros of Bessel functions. SIAM J. Math. Anal. 17, 1483–1488 (1986) · Zbl 0597.33007 · doi:10.1137/0517106
[7]Elbert, Á., Laforgia, A.: Further results on McMahon’s asymptotic approximations. J. Phys. A: Math. Gen. 33, 6333–6341 (2000) · Zbl 0962.33005 · doi:10.1088/0305-4470/33/36/305
[8]Gatteschi, L.: Valutazione dell’errore nella formula di McMahon per gli zeri della J n (x) di Bessel nel caso 0 n 1. Rivista Mat. Univ. Parma 1, 347–362 (1950)
[9]Gatteschi, L.: Funzioni Speciali. UTET, Torino (1973)
[10]Gatteschi, L.: Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144, 7–27 (2002) · Zbl 1008.65011 · doi:10.1016/S0377-0427(01)00549-0
[11]Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)
[12]Laforgia, A.: Sugli zeri delle funzioni di Bessel. Calcolo 17, 211–220 (1980) · Zbl 0464.33007 · doi:10.1007/BF02576701
[13]Lorch, L., Szego, P.: Higher monotonicity properties of certain Sturm-Liouville functions. Acta Math. 109, 55–73 (1963) · Zbl 0111.06502 · doi:10.1007/BF02391809
[14]Muldoon, M.E.: Continuous ranking of zeros of special functions. J. Math. Anal. Appl. (2008). doi: 10.1016/j.jmaa.2008.01.082
[15]Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)