Polynomials related to the Bessel functions
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- by F. T. Howard PDF
- Trans. Amer. Math. Soc. 210 (1975), 233-248 Request permission
Abstract:
In this paper we examine the polynomials ${W_n}(a)$ defined by means of \[ - 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{W_n}(a){x^n}/n!} .\] These polynomials are closely related to the zeros of the Bessel function of the first kind of index —3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of ${W_n}(a)$ are investigated. We show that if $n$ is even then ${W_n}(a) > 0$ for all $a$, and if $n$ is odd then ${W_n}(a)$ has only the one real root $a = 1/2$. Also we find upper and lower bounds for all $b$ such that ${W_n}(a + bi) = 0$. The problem of multiple roots is discussed and we show that if $n \equiv 0,1,5,8$ or 9 $(\bmod \; 12)$, then ${W_n}(a)$ has no multiple roots. Finally, if $n \equiv 0,1,2,5,6$ or 8 $(\bmod \; 12)$, then ${W_n}(a)$ has no factor of the form ${a^2} + ca + d$ where $c$ and $(\bmod \; 12)$ are integers.References
- John Brillhart, On the Euler and Bernoulli polynomials, J. Reine Angew. Math. 234 (1969), 45–64. MR 242790, DOI 10.1515/crll.1969.234.45
- John Brillhart, Some modular results on the Euler and Bernoulli polynomials, Acta Arith. 21 (1972), 173–181. MR 304298, DOI 10.4064/aa-21-1-173-181
- L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc. 14 (1963), 1–9. MR 166147, DOI 10.1090/S0002-9939-1963-0166147-X
- L. Carlitz, Recurrences for the Rayleigh functions, Duke Math. J. 34 (1967), 581–590. MR 214821, DOI 10.1215/S0012-7094-67-03463-1
- F. T. Howard, A property of a class of nonlinear difference equations, Proc. Amer. Math. Soc. 38 (1973), 15–21. MR 309849, DOI 10.1090/S0002-9939-1973-0309849-0
- F. T. Howard, Properties of the van der Pol numbers and polynomials, J. Reine Angew. Math. 260 (1973), 35–46. MR 318054, DOI 10.1515/crll.1973.260.35
- F. T. Howard, The van der Pol numbers and a related sequence of rational numbers, Math. Nachr. 42 (1969), 89–102. MR 258739, DOI 10.1002/mana.19690420107
- F. T. Howard, Generalized van der Pol numbers, Math. Nachr. 44 (1970), 181–191. MR 299552, DOI 10.1002/mana.19700440115
- G. S. Kazanzidis, On a congruence and on a practical method for finding the highest power of a prime $p$ which divides the binomial coefficient ${A\choose B}$, Bull. Soc. Math. Grèce (N.S.) 6 II (1965), no. fasc. 2, 358–360 (Greek). MR 207625
- Nand Kishore, A class of formulas for the Rayleigh function, Duke Math. J. 34 (1967), 573–579. MR 214820
- Nand Kishore, A structure of the Rayleigh polynomial, Duke Math. J. 31 (1964), 513–518. MR 164071
- Nand Kishore, The Rayleigh function, Proc. Amer. Math. Soc. 14 (1963), 527–533. MR 151649, DOI 10.1090/S0002-9939-1963-0151649-2
- Nand Kishore, The Rayleigh polynomial, Proc. Amer. Math. Soc. 15 (1964), 911–917. MR 168823, DOI 10.1090/S0002-9939-1964-0168823-2
- H. L. Krall and Orrin Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100–115. MR 28473, DOI 10.1090/S0002-9947-1949-0028473-1 D. H. Lehmer, Zeros of the Bessel function ${J_v}(x)$, Math. Comp. 1 (1943-1945), 405-407.
- N. Liron, A recurrence concerning Rayleigh functions, SIAM J. Math. Anal. 2 (1971), 496–499. MR 306573, DOI 10.1137/0502049
- N. Liron, Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half, Math. Comp. 25 (1971), 769–781. MR 304726, DOI 10.1090/S0025-5718-1971-0304726-X
- Lee Lorch, The limits of indetermination for Riemann summation in terms of Bessel functions, Colloq. Math. 15 (1966), 313–318. MR 206559, DOI 10.4064/cm-15-2-313-318 N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, New York, 1954.
- Robert Spira, The nonvanishing of the Bernoulli polynomials in the critical strip, Proc. Amer. Math. Soc. 17 (1966), 1466–1467. MR 203114, DOI 10.1090/S0002-9939-1966-0203114-4 B. van der Pol, Smoothing and “unsmoothing", Probability and Related Topics in Physical Sciences, New York, 1957, pp. 223-235.
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 210 (1975), 233-248
- MSC: Primary 10A40; Secondary 33A40
- DOI: https://doi.org/10.1090/S0002-9947-1975-0379348-5
- MathSciNet review: 0379348