Frank M. Davis is listed in the US CW Research Database as having enlisted as a Private in Company D, 2nd Cherokee Cavalry, IT, however, that listing seems to be the result of a file under his name which gives this intelligence:

"No record as been found in this office or service, capture or parole of a man named Frank M. Davis as a member of either the 2nd Reg't. Cherokee Mounted Volunteers (also called Rifles), or the 1st (also called the 2nd) Reg't Cherokee Mounted Volunteers, Confederate States Army."
US Army Adjutant General to Oaklahoma State Board of Pension Commissioners, 12/01/1915

He may be one of the two men recorded below.

George Martin

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Frank M. Davis, enlisted as a Sergeant
"F" Co. AR Dawon's-Hardy's Infantry

This regiment's records have not been in Footnote at this time.

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Frank M. Davis, age 25 year old Farmer, resident of Cooper County, Missouri, enlisted as a Private in Company A, 2nd Missouri Cavalry, January 14, 1862 at Springfield, Missouri, surrendered at Citronelle, Alabama May 4, 1865, paroled at Columbus, Mississippi May 16

This company originally belonged to the Missouri State Guard, having joined that organization in May 1861

There were six other men with the surname Davis also in Company A

See also:

http://members.tripod.com/2ndmocavcsa/short_history.htm
http://findarticles.com/p/articles/mi_qa3905/is_200603/ai_n17176545/