Can I write PostgreSQL functions on Ruby on Rails?

This part of your question:

I know we can create it manually in PostgreSQL, but the "magic" with Active Record is that the database can be recreated with all the models.

tells me that you're really looking for a way to integrate PostgreSQL functions with the normal Rails migration process and Rake tasks such as db:schema:load.

Adding and removing functions in migrations is easy:

def up
  connection.execute(%q(
    create or replace function ...
  ))
end

def down
  connection.execute(%q(
    drop function ...
  ))
end

You need to use separate up and down methods instead of a single change method because ActiveRecord will have no idea how to apply let alone reverse a function creation. And you use connection.execute to feed the raw function definition to PostgreSQL. You can also do this with a reversible inside change:

def change
  reversible do |dir|
    dir.up do
      connection.execute(%q(
        create or replace function ...
      ))
    end
    dir.down do
      connection.execute(%q(
        drop function ...
      ))
    end
  end
end

but I find that noisier than up and down.

However, schema.rb and the usual Rake tasks that work with schema.rb (such as db:schema:load and db:schema:dump) won't know what to do with PostgreSQL functions and other things that ActiveRecord doesn't understand. There is a way around this though, you can choose to use a structure.sql file instead of schema.rb by setting:

config.active_record.schema_format = :sql

in your config/application.rb file. After that, db:migrate will write a db/structure.sql file (which is just a raw SQL dump of your PostgreSQL database without your data) instead of db/schema.rb. You'll also use different Rake tasks for working with structure.sql:

• db:structure:dump instead of db:schema:dump
• db:structure:load instead of db:schema:load

Everything else should work the same.

This approach also lets you use other things in your database that ActiveRecord won't understand: CHECK constraints, triggers, non-simple-minded column defaults, ...

If your only requirement is creating them somewhere in your Rails app, this is possible through ActiveRecord::Base.connection.execute, which you can use to execute raw SQL queries.

stmt = 'CREATE FUNCTION...'
ActiveRecord::Base.connection.execute stmt

You would then call the function using ActiveRecord::Base.connection.execute as well (I'd imagine you'd have methods in your model to handle this).