Father is Hamilton Allison, this was proven by DNA testing. Jasper and his brother John took the last name of his mother first husband after Hamilton Allison died.
Gravestone and Death Cert. both state birth year as 1883. This is not possible as Jasper father only married his mother in 1891. Most documents from the early 1900s Jasper states that his birth year was 1894-1895. I don't know why he changed his birth year in his later years of his life.
Sources
"United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M3JX-PF4 : accessed 16 January 2019), Jasper Allison in household of Mary Allison, Militia Districts 1051,838, 1244, Wahoo, Shoal Creek, Frogtown, Lumpkin, Georgia, United States; citing enumeration district (ED) 98, sheet 2A, family 115, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, 1972.); FHL microfilm 1,240,209.
"United States Census, 1910," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MLLR-JPF : accessed 16 January 2019), Jasper W Ledford in household of Mary E Ledford, Shoal Creek, White, Georgia, United States; citing enumeration district (ED) ED 156, sheet 15B, family 271, NARA microfilm publication T624 (Washington D.C.: National Archives and Records Administration, 1982), roll 217; FHL microfilm 1,374,230.
"United States Census, 1930," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:33N8-J2M : accessed 16 January 2019), Jasper Ledfort, Moultrie, Colquitt, Georgia, United States; citing enumeration district (ED) ED 4, sheet 12B, line 62, family 276, NARA microfilm publication T626 (Washington D.C.: National Archives and Records Administration, 2002), roll 348; FHL microfilm 2,340,083.
"United States Census, 1940," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:K7LZ-9S3 : 16 March 2018), Jasper Ledford, Militia District 799, Colquitt, Georgia, United States; citing enumeration district (ED) 35-1, sheet 8B, line 54, family 121, Sixteenth Census of the United States, 1940, NARA digital publication T627. Records of the Bureau of the Census, 1790 - 2007, RG 29. Washington, D.C.: National Archives and Records Administration, 2012, roll 660.
"United States World War I Draft Registration Cards, 1917-1918," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:KZZH-QQX : 13 March 2018), Jasper Ledford, 1917-1918; citing Colquitt County, Georgia, United States, NARA microfilm publication M1509 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,557,018.
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DNA Connections
It may be possible to confirm family relationships with Jasper by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: