A STUDY OF THE LAGUERRE POLYNOMIALS

Abstract

In this article the complex Laguerre polynomials L(α−k)(z) are presented, together with some expressions that result from k

them, which appear using definitions mentioned in the text. In addition, a review of the properties of the Laguerre poly-

nomials and their convergence in mean, studied by various authors throughout history, is carried out. The convergence of

the Laguerre polynomials begins with the studies of Pollard, who posits that for it to be fulfilled

Rb PN p α limN−→∞ a f(x)− 0 anPn(x) dF(x) = 0,thenp = 2,thenAskeyandWaingerposeaLaguerrefunctionLn(x) =

xα/2rn exp(−x/2)Lαn(x) which converges if 4/3 < p < 4. In the following investigation, Muckenhoupt indicates

that in the Laguerre polynomials, the terms will not converge to 0 at the mean if p is not between 4/3 and 4, but

this time proving that p is a fixed number that satisfies 1 ≤ p ≤ 4/3 or 4 ≤ p ≤ ∞. Then the same Mucken-

houpt generalizes Askey and Wainger’s results for mean convergence with 1 < p < ∞. The results improve in terms

of weighting functions, their research is based on inequalities that required a larger weighting function on the right

sidethanontheleftR∞|Sα(f,x)U(x)|pdx ≤ CR∞|f(x)V(x)|pdx. Later,Poianiprovesinequalitiesoftheform 0n0

∥σn(f,x)W(x)∥p ≤ ∥f(x)W(x)∥p where σn is the nth (C,1) convergence of the Laguerre series of f, W(x) the weight function of a particular form and the norm Lp is taken over (0, ∞), here only a weight function is used. Finally, there is the research carried out by Mario Riera, who studies said convergence with Dirac deltas, in this case for Laguerre with a delta at zero