The two airline transport pilots were conducting a full-stop landing at their home airport. The airplane was put on a 5 to 8 mile final as the pilot read and quickly complete the checklist. The airplane was configured with full flaps and when the pilot went to actuate the landing gear he diverted his attention to his deteriorating airspeed and glidepath deviation. The pilot made the necessary correction to stabilize the approach. At 500 feet above the ground, the pilot glanced over to the gear indicator and thought he saw "three down." The pilot continued the approach and the airplane landed with the gear in the retracted position. Neither pilot reported hearing the gear warning horn.