We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that L n (x) = Pn j=0 `n− j+r n− j ́ x / j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950’s by Grosswald and proven recently by Filaseta and T...

The idea of looking at the prime factorization of the coefficients of a polynomial in Z[x] in order to establish its irreducibility (over Q) goes back to the classical Schönemann-Eisenstein criterion first derived in [29] and [6] in the middle of the 19th century. At the beginning of the 20th century, G. Dumas [5], again making use of primes that divide the coefficients of a polynomial, introdu...

Let K be a number field, and let λ(x, t) ∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of α ∈ K for which the specialized polynomial λ(x, α) is K-reducible. We apply this to show that for any fixed n ≥ 10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L (t) n (x) are K-irreduci...

Motivated by an expression Persson and Strang on integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre Gegenbauer polynomials as well original polynomial with even index.

In this paper, we use a digital laser to generate high-radial-order Laguerre-Gaussian, LGp,0 modes by loading digital holograms on a phase-only spatial light modulator that act as an end mirror of a diode-end-pumped laser resonator. The digital holograms were encoded with an amplitude ring mask, which contained absorption rings that match the p-zeros of the Laguerre polynomial. We demonstrate t...

In this survey the role of Jacobi’s last multiplier in mechanical systems with a position dependent mass is unveiled. In particular, we map the Liénard II equation ẍ + f(x)ẋ + g(x) = 0 to a position dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the von Roos ordering technique. Finally we show ...

Abstrakt This article deals with the use of the Laguerre functions in the data compression. After the short introduction the definition of the Laguerre polynomials and functions is given. The application of the discrete Laguerre transform on the data compression is shown. It is pointed out that the discrete Laguerre transform can give better results than the discrete cosine transform in the tas...