On the analyticity of Laguerre series

Abstract

The transformation of a Laguerre series f(z) = ∑∞n=0λ(α)nL(α)n(z) to a power series f(z) = ∑∞n=0γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients λ(α)n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ(α)n either decay exponentially or factorially as n → ∞. The situation is much more complicated—but also much more interesting—if the λ(α)n decay only algebraically as n → ∞. If the λ(α)n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the λ(α)n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients λ(α)n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series p+1Fp, but if the λ(α)n have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations—in particular the so-called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189–371 (equation (8.4-4)))—are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369–71, Guseinov 2001 Int. J. Quantum Chem. 81 126–29, Guseinov 2002 Int. J. Quantum Chem. 90 114–8) is investigated.