If you're talking about a true Burnside carbine with serial number 17916, and not a Burnside-made Spencer, my resources don't show a match but show the following as the three closest numbers:
17895 held by Co. C, 12th New York Volunteer Cavalry, on September 15, 1863
17903 held by Co. I , 4th Wisconsin Volunteer Cavalry, in August 1864 and on October 1, 1864 (two entries)
17914 held by Co. C, 12th New York Volunteer Cavalry, on September 15, 1863
Most of the Burnsides for which I have info that fall in the serial number range between 16,500 through 19,000 are listed with the 12th New York Volunteer Cavalry or the 4th Wisconsin Volunteer Cavalry.
I adopt Two Flints' serial number disclaimer--the serial numbers I provided may mean nothing other than some of the Burnside carbines made at about the same time as yours were in the possession of the listed units. Since I am relying on research done by others and have not verified the original sources, I cannot truly vouch for any of the information.
--DJ