January 0

January 0 does not exist in any calendars. It sometimes refers to December 31.
In an ephimeris.
January 0 also refers to the day before January 1 in an annual ephemeris. It keeps the date in the year for which the ephemeris was published. It avoids any reference to the previous year, even though it is the same day as December 31 of the previous year.
Software.
In Microsoft Excel, the epoch of the 1900 date format is January 0, 1900.