Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

Angelo B. Mingarelli ()
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Angelo B. Mingarelli: School of Mathematics and Statistics Carleton University, Ottawa, ON K1S 5B6, Canada

Mathematics, 2023, vol. 11, issue 2, 1-19

Abstract: We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ ( 3 ) and ζ ( 2 ) in terms of Apéry’s now classic sequences.

Keywords: irrational numbers; three term recurrence relations; principal solution; dominant solution; four term recurrence relations; Apéry; Riemann zeta function; difference equations; asymptotics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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