Is it necessary to ANALYZE a table after an index has been created?
If the index is just on simple columns like in your case, it is not necessary to ANALYZE the table after you create the index.
That is because statistics on the value distribution of the table columns are always collected, no matter if there is an index on the column or not.

However, if you are indexing an expression like upper(some_column) or (CAST(some_column AS date)), you should run ANALYZE after creating the index.
PostgreSQL will then also collect statistics on the value distribution of the indexed expression. This happens automatically whenever autoanalyze runs, but it is a good idea to do it manually right after creating the index so you have good statistics right away.