University of Kashan Mathematics Interdisciplinary Research 2538-3639 11 2 2026 06 01 k-Fibonacci and k-Lucas Differential Equations with New Spirals 161 176 115536 10.22052/mir.2026.257640.1541 EN Mehmet Pakdemirli ‎Mechanical Engineering Department, Manisa Celal Bayar University, ‎Muradiye‎, ‎Yunusemre‎, ‎Manisa‎, ‎Turkey İhsan Dolapci ‎Mechanical Engineering Department, Manisa Celal Bayar University, ‎Muradiye‎, ‎Yunusemre‎, ‎Manisa‎, ‎Turkey Journal Article 2025 09 22 ‎Fibonacci sequences and the spirals formed by employing them have found vast applications in creations and natural phenomena‎. ‎In this study‎, ‎new k-Fibonacci and k-Lucas differential equations are proposed‎. ‎First‎, ‎the k-Fibonacci and k-Lucas sequences are expressed as difference-differential equations‎. ‎Then‎, ‎from the difference-differential equations‎, ‎the associated continuous differential equations are derived‎, ‎which are linear second-order differential equations‎. ‎The initial conditions for the differential equations are written with inspiration from the k-Fibonacci and k-Lucas sequences‎. ‎The solutions‎, ‎which are new spirals‎, ‎are expressed in polar form‎. ‎The spirals produce approximately the k-Fibonacci and k-Lucas numbers at constant steps of angular displacements‎. https://mir.kashanu.ac.ir/article_115536_7cb311a2267d6aab13c8cd60b661f84b.pdf