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Transactions of the American Mathematical Society

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Polynomials related to the Bessel functions


Author: F. T. Howard
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
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Abstract: In this paper we examine the polynomials $ {W_n}(a)$ defined by means of

$\displaystyle - 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum\limits_{n = 0}^\infty {{W_n}(a){x^n}/n!} .$

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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0379348-5
Keywords: Rayleigh function, Bernoulli polynomial, Euler polynomial, van der Pol polynomial, Bessel polynomials, polynomials $ \pmod 2$
Article copyright: © Copyright 1975 American Mathematical Society

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