The obituary names a spouse, but I have not found anything to prove he is Joe’s father.
commented by George Fulton
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I would actually question if such a person ever existed. In 1900 she was Lucinda J Jackson, unmarried at age 36, living with her parents. In 1910 she was Lucinda J Jackson, a Widow, aged 46, with a son born in 1903.
That's not a very large window (1900 to 1903) to marry, give birth, and have the husband die. (Desertion, I could see.)
In 1930, her name was now Louise J Jackson, Divorced, aged 66, living with the son from the 1910 census (so it's the same person, just with a first name spelling difference).
It seems she went more by Jane, as she also is in records — and that obit (which is viewable as an image attached to a profile on an ancestry tree) — as L. Jane Jackson.
A fictional "husband" of the same last name would help to explain (and prevent the ostracism that went with unwed motherhood) the same last name before and after the birth of the child, especially for a church-going woman who was, by her death, a highly respected member of her community.
The full text of that obit is —
Mrs. Jane Jackson.
Dunn.—Mrs. L. Jane Jackson, 73, died at the home of her son, Joe Turner Jackson, on Lillington, Route 2, Tuesday afternoon at 6:30 o'clock following an illness of 6 weeks.
Funeral services were held Wednesday afternoon at the Grove Presbyterian Church. Burial was in the church cemetery.
Mrs. Jackson, a daughter of the late Y. S. and Sybil Norris Jackson, was a native of Sampson County, and the widow of Allen Jackson. She was a charter member of the Grove Presbyterian Church and was well known and highly respected in her community.
Surviving are one son, Joe T. Jackson of Lillington Route 2; a brother, Nathan T. Jackson of Raleigh and two sisters, Mrs. Mary Jones of Dunn and Mrs. Ellen Jones of Lillington, Route 2.