Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

Abstract

Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1(a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter c of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1(a, b; c; z) are derived.

Fourier transform of hydrogen-type atomic orbitals

“…Guseinov's original notation (68) does not allow nonintegral values of α. In order to rectify this obvious deficiency, I introduced in [23,85] the alternative definition (70) which uses the modern mathematical notation for the generalized Laguerre polynomials.…”

“…Guseinov's original definition (68) implies that his functions are according to [84,Eq. (4)] orthogonal with respect to the weight function w(r) = [n ′ /(ζr)] α [84, Eq.…”

“…In his later articles [86,87], Guseinov generalized his so-called frictional quantum number from originally α = 1, 0, −1, −2, · · · to α ∈ (−∞, 3), which corresponds to k ∈ (−3, ∞) in my notation. This change could not be done with Guseinov's original definition (68). Therefore, Guseinov finally had to use the modern mathematical notation for his functions (compare [86, Eqs.…”

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

“…Chatterjee and Roy [20] consider a modification of standard Levin-type methods tailored to the hypergeometric function, and Gautschi [27] considers evaluation of both Gaussian and confluent hypergeometric functions for complex arguments, but real parameters, using Gaussian quadrature to evaluate integral representations of the functions. Weniger looked at using traditional series acceleration methods but irregular input data in [58] and considered divergent hypergeometric series at z = −1 using a method tailored to alternating series in [60]. Finally Kalmykov [34] and Kalmykov et al [35] consider an expansion of the Gaussian function near integer values of its parameters.…”

On the analyticity of Laguerre series

“…As I had shown in several articles [84,85,136,137,139,140,143,147,148,155,162], nonlinear sequence transformations can be extremely useful in this respect.…”