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Inequalities and monotonicity properties for zeros of Hermite functions

Published online by Cambridge University Press:  14 November 2011

Árpád Elbert  and
Martin E. Muldoon
Árpád Elbert
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary
Martin E. Muldoon
Affiliation:
Department of Mathematics, York University, Toronto, Ontario, Canada, M3J 1P3
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Extract

We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each non-negative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1, and that each zero increases from – ∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic function of λ. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of Hλ(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 57 - 75
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Abramowitz, M. and Stegun, I. A. (eds). Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards, Applied Mathematics Series, vol. 55 (Washington, DC, 1964).Google Scholar
2Appell, P.. Sur les transformations des équations différentielles linéaires. C. R. Acad. Sci. Paris 91 (1880), 211–214.Google Scholar
3Bochner, S.. Completely monotone functions of the Laplace operator for torus and sphere. Duke Math. J. 3 (1937), 499–502.CrossRefGoogle Scholar
4Durand, L.. Nicholson-type integrals for products of Gegenbauer functions and related topics. In Theory and applications of special functions (ed. Askey, R.), pp. 353–374 (Academic, 1975).CrossRefGoogle Scholar
5Durand, L.. Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Analysis 9 (1978), 76–86.CrossRefGoogle Scholar
6Elbert, Á.. Concavity of the zeros of Bessel functions. Studia Sci. Math. Hungar. 12 (1977), 81–88.Google Scholar
7Elbert, Á and Laforgia, A.. On the square of the zeros of Bessel functions. SIAM J. Math. Analysis 15 (1984), 206–212.CrossRefGoogle Scholar
8Elbert, Á. and Laforgia, A.. On the convexity of the zeros of Bessel functions. SIAM J. Math. Analysis 16 (1985), 614–619.CrossRefGoogle Scholar
9Elbert, Á. and Laforgia, A.. Further results on the zeros of Bessel functions. Analysis 5 (1986), 71–86.Google Scholar
10Elbert, Á. and Muldoon, M. E.. On the derivative with respect to a parameter of a zero of a Sturm–Liouville function. SIAM J. Math. Analysis 25 (1994), 354–364.CrossRefGoogle Scholar
11Erdélyi, A., Magnus, W., Oberhettingen, F. and Tricomi, F. G.. Higher transcendental functions, vol. 1 (New York: McGraw-Hill, 1953).Google Scholar
12Erdélyi, A., Magnus, W., Oberhettingen, F. and Tricomi, F. G.. Higher transcendental functions, vol. 2 (New York: McGraw-Hill, 1954).Google Scholar
13Friedland, S. and Hayman, W. K.. Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51 (1976), 133–161.CrossRefGoogle Scholar
14Gradshteyn, I. S. and Ryzhik, I. M.. Tables of integrals, series and products, 4th edn (Academic, 1965).Google Scholar
15Hartman, P.. Ordinary differential equations, 3rd edn (Birkhäuser, 1982).Google Scholar
16Hayman, W. K. and Ortiz, E. L.. An upper bound for the largest zero of Hermite's function with applications to subharmonic functions. Proc. R. Soc. Edinb. A 75 (1975–1976), 183–197.CrossRefGoogle Scholar
17Ismail, M. E. H. and Zhang, R.. On the Hellmann–Feynman theorem and the variation of zeros of certain special functions. Adv. Appl. Math. 9 (1988), 439–446.CrossRefGoogle Scholar
18Ismail, M. E. H. and Zhang, R.. On the Hellman–Feynman theorem and the variation of zeros of special functions. In Ramanujan Int. Symp. on Analysis (Pune, 1987) (ed. Thakare, N. K., Sharma, K. C. and Raghunathan, T. T.), pp. 151–183 (New Delhi: Macmillan of India, 1989).Google Scholar
19Leighton, W. and Morse, M.. Singular quadratic functionals. Trans. Am. Math. Soc. 40 (1936), 252–86.Google Scholar
20Lewis, J. T. and Muldoon, M. E.. Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Analysis 8 (1977), 171–178.CrossRefGoogle Scholar
21Lorch, L., Muldoon, M. E. and Szego, P.. Higher monotonicity properties of certain Sturm–Liouville functions. III. Can. J. Math. 22 (1970), 1238–1265.CrossRefGoogle Scholar
22Schoenberg, I. J.. Metric spaces and completely monotone functions. Ann. Math. (2) 39 (1938), 811–841.CrossRefGoogle Scholar
23Lorch, L. and Newman, D. J.. A supplement to the Sturm separation theorem, with applications. Am. Math. Monthly 72 (1965), 359–366; Acknowledgment of priority, 980.CrossRefGoogle Scholar
24Richardson, R. G. D.. Contribution to the study of oscillation properties of the solutions of linear differential équations of the second order. Am. J. Math. 40 (1918), 283–316.CrossRefGoogle Scholar
25Schläfli, L.. Über die Convergenz der Entwicklung einer arbiträren Function f(x) nach den Bessel'schen Functionen Jα(β1x), J α(β2x), Jα(β3x),…, wo β 1,β 2,β 3,… die positiven Wurzeln der Gleichung Jα(β) vorstellen. Math. Ann. 10 (1876), 137–142.CrossRefGoogle Scholar
26Szegő, G.. Orthogonal polynomials, vol. 23, 4th edn (AMS Colloquium Publications, 1975).Google Scholar
27Watson, G. N.. A treatise on the theory of Bessel functions, 2nd edn (Cambridge University Press, 1944).Google Scholar