A unified approach to derive explicit solutions of generalized second-order linear recurrences and applications

We illustrate the versatility of our previous work on closed-form solutions of second-order linear recurrences by specializing the underlying counting sets to prove new identities. Extensions allowing the sequence values to be functions of a variable t as well as the integer variable n are mentioned and applied to derive closed-form solutions, both classical and new. Connections with continued fractions are established and used to settle several conjectures from the Ramanujan Machine. Finally, alternative verifications of well-known continued fractions such as those of zeta functions and continued fractions in Ramanujan's second notebook are presented. (c) 2023 Published by Elsevier B.V.