A little bit of follow-up:
1) Thanks to Sherri Bobish for mentioning my find of a published notice of my grandfather's court-approved name change in a newspaper from 1940
(I clicked on that link and searched for "Jay Joseph Kent" -- and got nothing) ;
2) I'm still not clear what happened years later (legally or otherwise) to cause the New York City Department of Health to amend the birth certificate quite a few years later : I don't know if he (or a lawyer) communicated that he was only now using his new name.
(Manhattan telephone books from earlier in the 1940s -- during US involvement in World War II -- accessed via Ancestry.com -- indicate that he was still publicly known as "Joseph Kantor" -- until he decided not to be known by that name any more.
And I do not know if his first wife -- my grandmother -- ever used "Kent" in her name.)
Ethan W. Kent in New York City.
PS: It was only after he died (in 1992) that I learned (possibly my father had not known either) that my grandfather's changed "first" name was "Jay": he was known to me and my brother and sister as "Grandpa Joe", and while his return-address (postal, of course) mailing labels read "J. Joseph Kent", they said nothing about "Jay" -- and I got the impression (I think from my father) that "J." was just an affectation -- and referred to the "J." in "Joseph".
It is my guess (especially after learning from censuses from his childhood that census-takers were told his name was "Jacob") that perhaps "Jay" alluded to a Hebrew name of Yaakov Yosef.
E. W. K.