The irreducibility of the Bessel polynomials

Filaseta, Michael; Trifonov, Ognian

doi:10.1515/crll.2002.069

Cited by 21 publications

(43 citation statements)

References 7 publications

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“…The techniques used for these results will be similar to those used in [6] and [7]. The above three results show that for 0 < |a n | < M , f (x) is irreducible except possibly for finitely many pairs (a n , n).…”

Section: Introductionmentioning

confidence: 80%

A generalization of a second irreducibility theorem of I. Schur

Allen

Filaseta

2003

Acta Arith.

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Section: Introductionmentioning

confidence: 80%

A generalization of a second irreducibility theorem of I. Schur

Allen

Filaseta

2003

Acta Arith.

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“…Recent work of a similar nature has been done by Filaseta [1,2] and by Filaseta and Trifonov [3]. We note also that the above results can be made effective so that for any fixed α ∈ Q, α not a negative integer, it is possible to determine a finite set S = S(α) of m such that the polynomial in Theorem 2 is irreducible (for a j as stated there) provided m ∈ S.…”

mentioning

confidence: 58%

On the irreducibility of the generalized Laguerre polynomials

Filaseta

Lam

2002

Acta Arith.

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Let m ≥ 1 and a m be integers. Let α be a rational number which is not a negative integer such that α = u v with gcd(u, v) = 1, v > 0. Let φ(x) belonging to Z[x] be a monic polynomial which is irreducible modulo all the primes less than or equal to vm + u. Let a i (x) with 0 ≤ i ≤ m − 1 belonging to Z[x] be polynomials having degree less than deg φ(x). Assume that the content of (a m a 0 (x)) is not divisible by any prime less than or equal to vm + u. In this paper, we prove that the polynomialsb j a j (x)φ(x) j ) are irreducible over the rationals for all but finitely many m, where bis irreducible over rationals for each α ∈ {0, 1, 2, 3, 4} unless (m, α) ∈ {(1, 0), (2, 2), (4, 4), (6, 4)}. For proving our results, we use the notion of φ-Newton polygon and some results from analytic number theory. We illustrate our results through examples.

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“…We will make use of the following estimate, which can be found in [8], for π(x), the number of primes ≤ x. The first result in Lemma 3 with intervals (x, 1.01x] is obtained in [6] by making direct use of estimates like Lemma 2 above from [8]. A proof of the second result follows by noting [1.05 · 213] = 223 is prime and by simply checking the lengths of the gaps between primes in the interval [223,2503] (the number 2503 is the smallest prime > 2479).…”

Section: Preliminary Materialmentioning

confidence: 99%

“…The conditions on a n were different for these results, but in a manner similar to that just described, the results there were best possible. The general techniques used for establishing Theorem 1 are similar to those used in [1], [5] and [6]. The authors were not, however, able to take advantage of work by E. F. Ecklund, Jr., R. B. Eggleton, P. Erdős, and J. L. Selfridge [3] that played a crucial role in the prior two papers [5] and [1] on the subject.…”

Section: Introductionmentioning

confidence: 99%