We introduce various sum formulae involving products of two Fibonomial/Lucanomial coefficients or quotients of two such coefficients over a third one. In order to prove the claims we will convert them into their q-forms and then use known terminating well-poised q-series identities.

Stick-slip in drilling happens when the drill bit and assembly repeatedly stop and slide due to factors like friction and torque. Using better lubricants in the drilling mud can help reduce friction, lessening stick-slip issues. Nanotechnology has improved drilling mud, making it more resistant to heat and enhancing its viscosity and friction properties. This research extensively investigates the use of nanofluids in drilling processes through thorough lab tests. It examines the impact of introducing NanoSiO2 into two types of drilling mud—freshwater bentonite (FWB) and polymer-based mud—at concentrations of 0.04%, 0.08%, 0.15%, and 0.31% under higher temperatures. The study also explores two methods of dispersing NPs using ultrasonic and solvent techniques. The results show that SiO2 nanoparticles notably improve the flow and filtering abilities of FWB mud but have negative effects on polymer-based mud. A 0.05% addition of NanoSiO2 significantly reduces friction by 52% in FWB mud. In FWB mud, all NanoSiO2 concentrations lead to a 25% decrease in mud cake thickness. Furthermore, adding NanoSiO2 to FWB mud decreases torque, with the most substantial reduction of 40% observed at 0.09% concentration (0.35 gm). On the contrary, incorporating NanoSiO2 into polymer-based mud increases fluid loss volume for all concentrations, reaching up to 36% compared to the base mud. Additionally, filter cake thickness increases eightfold, while lubricity factor doubles. This study offers insights into the advantages of using SiO2 nanoparticles in drilling mud and their impact on the fluid properties during drilling.