How to create an index in Postgres on an expression involving TEXT columns and a DATE column, concatenated?

I want to be able to search for people "similar" to what I have in the database (e.g. a person with a typo in their name or date of birth). I want to use the levenshtein function for that in the following way:

create temp table person (first_name text, last_name text, dob date);
select * from person where levenshtein('johndoe2000-01-01', lower(first_name || last_name || dob)) < 4;

(I am aware this can be probably optimized a bit by e.g. using YYYYMMDD instead of YYYY-MM-DD, but that is irrelevant to my question.)

I thought that I could speed up the search with an index. However, I don't know how to create one that would work:

create index fuzzy on person ((lower(first_name || last_name || dob)));

results in ERROR: functions in index expression must be marked IMMUTABLE. I tried a few things: using concat instead of ||, casting dob to text explicitly or with to_char, but to no avail. What am I doing wrong? It seems that the problem results from mixing text and date – I can create an index with only text columns or only the date column without a problem.

The exercise is pointless: you can never use an index to speed up a query that uses levenshtein().

Casting dates to strings depends on database setting that might change, and so it is not immutable.

select '2021-04-03'::date::text;
 2021-04-03
set datestyle TO 'german';
select '2021-04-03'::date::text;
 03.04.2021

You could solve that problem in a variety of ways. But once you do, it still wouldn't work as levenshtein distance is generally not indexable.