Expansions in Laguerre polynomials of negative order (Q1899347)

Let \(n\) be a non-negative integer, \(-(n+2)< \alpha<- (n+1)\). Let \(f(x)\) be a given function, and suppose there exists a polynomial \(P_n (x)\) of degree \(n\) such that \[ \int_0^\infty |f(x)- P_n (x)|x^\alpha dx<\infty. \] The Hadamard's finite part (f.p.) of the infinite integral \(\int_0^\infty f(x) x^\alpha dx\) is defined by \[ \text{f.p. }\int_0^\infty f(x) x^\alpha dx= \int_0^\infty |f(x)- P_n(x) |x^\alpha dx. \] This paper is concerned with the Laguerre expansions of non-integer \(\alpha< -1\) defined by \[ f(x)\cong \sum^\infty_{k=0} a_k L_k^\alpha (x), \] where \(L_k^\alpha (x)\) is the \(k\)-th Laguerre polynomial of order \(\alpha\) and \[ a_k= {{k!} \over {\Gamma(k +\alpha+1)}} \text{ f.p. }\int_0^\infty f(y) L_k^\alpha (y) dy. \] The author proves the pointwise convergence of expansions of functions for which the difference of the function and a suitable polynomial satisfies a certain integrability condition.