41 research outputs found
Irreducibility of generalized Hermite-Laguerre Polynomials III
For a positive integer and a real number , the generalized
Laguerre polynomials are defined by \begin{align*}
L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots
(j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These orthogonal polynomials are
solutions to Laguerre's Differential Equation which arises in the treatment of
the harmonic oscillator in quantum mechanics. Schur studied these Laguerre
polynomials for its interesting algebraic properties. He obtained
irreducibility results of and and derived that the Hermite polynomials and
are irreducible for each . In this article, we
extend Schur's result by showing that the family of Laguerre polynomials
and with , where is the denominator
of , are irreducible for every except when where we
give the complete factorization. In fact, we derive it from a more general
result.Comment: Published in Journal of Number Theor
Stolarsky's conjecture and the sum of digits of polynomial values
Let denote the sum of the digits in the -ary expansion of an
integer . In 1978, Stolarsky showed that He conjectured that, as for , this limit
infimum should be 0 for higher powers of . We prove and generalize this
conjecture showing that for any polynomial with and and any base , For any we
give a bound on the minimal such that the ratio . Further, we give lower bounds for the number of such that
.Comment: 13 page
The sum of digits of and
Let denote the sum of the digits in the -ary expansion of an
integer . In 2005, Melfi examined the structure of such that . We extend this study to the more general case of generic and
polynomials , and obtain, in particular, a refinement of Melfi's result.
We also give a more detailed analysis of the special case , looking
at the subsets of where for fixed .Comment: 16 page
Perfect powers in products of terms of elliptic divisibility sequences
Diophantine problems involving recurrence sequences have a long history and
is an actively studied topic within number theory. In this paper, we connect to
the field by considering the equation \begin{align*} B_mB_{m+d}\dots
B_{m+(k-1)d}=y^\ell \end{align*} in positive integers with
and , where is a fixed integer and
is an elliptic divisibility sequence, an important class
of non-linear recurrences. We prove that the above equation admits only
finitely many solutions. In fact, we present an algorithm to find all possible
solutions, provided that the set of -th powers in is given. (Note
that this set is known to be finite.) We illustrate our method by an example.Comment: To appear in Bulletin of Australian Math Societ
On members of Lucas sequences which are products of factorials
Here, we show that if is a Lucas sequence, then the
largest such that with satisfies . We also give better bounds in case
the roots of the Lucas sequence are real