N.Y. Pub. Off. Law § 5

§ 5. Holding over after expiration of term. Every officer except a\njudicial officer, a notary public, a commissioner of deeds and an\nofficer whose term is fixed by the constitution, having duly entered on\nthe duties of his office, shall, unless the office shall terminate or be\nabolished, hold over and continue to discharge the duties of his office,\nafter the expiration of the term for which he shall have been chosen,\nuntil his successor shall be chosen and qualified; but after the\nexpiration of such term, the office shall be deemed vacant for the\npurpose of choosing his successor. An officer so holding over for one or\nmore entire terms, shall, for the purpose of choosing his successor, be\nregarded as having been newly chosen for such terms. An appointment for\na term shortened by reason of a predecessor holding over, shall be for\nthe residue of the term only.\n