Year: 2025
Type: article
Abstract: In this paper, we show how Zeilberger's algorithm can be used to find a closed-form expression for the sum of series containing a parameter. Series of the following form: ∑k=0∞(12−s)k(12+s)kk!21(a±k+n)iand∑k=0∞(12−s)k(12+s)kk!21((a±k)n)i,with i = 1, 2, 3 and with (c)n in the Pochhammer notation, provide a natural extension of a family of series that Ramanujan expressed using the Landau constants. They are ideally suited for applying Zeilberger's algorithm. Zeilberger's algorithm finds a linear recurrence for a sum of hypergeometric terms. By choosing these terms wisely, the resulting recurrence can sometimes be solved in closed form. As an example, we give a solution to an open problem in a 2022 article by Stewart that uses a series of this type.
Author: Paul Levrie
Institutions: University of Antwerp, KU Leuven
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416066492