Falcon 50EX pilot reported landing gear indication system malfunctioned during cruise. Upon arrival at the destination airport; the landing gear did not extend. The flight crew used the alternate gear extension procedure to extend the landing gear and continued to landing

Narrative

We were conducting a Part 91 repositioning flight from ZZZ1 to ZZZ. We departed in the mid afternoon from ZZZ1; and everything was uneventful until about an hour into our flight. While level at Flight Level XXX; we heard a bit of a low humming nose that seemed to come from the front of the aircraft. Shortly after it began the noise disappeared. At about the same time our landing gear handle started flashing red. The landing gear position panel didn't show anything out of the ordinary; and all lights tested normal. We contacted our maintenance personal and advised them of our issue. They conferred with Dassault technical support and told us that in all likelihood we seemed to have an issue with one of our proximity switches associated with our gear uplock. Since there was no other indications of any problems; and we were told this was most likely an indication failure we elected to continue our flight to our destination airport (also our home base). When we began our final approach to the ZZZ airport; we began configuring for landing and selected the gear handle to the down position. Once selected down; the landing gear handle light stopped flashing; however the landing gear didn't extend. We had no red lights; nor green down and locked indications; and no noise or sensations that the gear had extended. We advised the ZZZ final controller that we were having an issue with our landing gear; and requested vectors to try and troubleshoot. The controller asked us what altitude we would like for our vectors. I advised 'Either 4 or 5 feet'. We were given climb instructions; but we weren't able to hear precisely what altitude were supposed to fly. We requested clarification as we were at approximately 4;400 feet; and the controller advised us to maintain 4;000. We acknowledged the altitude; and advised that we had climbed through 4; but were returning to that altitude. Following the checklist; we initiated the alternate gear extension procedure. Upon completion; we had normal (for that condition) gear down and locked indications. That being three green; and two red lights for the inboard main gear doors. We advised ZZZ approach that we had a positive indication that our landing gear was extended; however; out of an abundance of caution; we did request that the airport have the equipment standing by for our arrival. We informed the approach controller and tower; that it was our intention to perform a normal landing; and to clear the runway and taxi to the FBO without assistance. The landing was uneventful; we then cleared the runway and proceeded to our FBO without further incident.At this time we are not sure what caused this landing gear abnormality. Our maintenance personal are trying to troubleshoot the issue with the assistance of the aircraft manufacturer.