How can I get a list of all functions stored in the database of a particular schema in PostgreSQL?

\df <schema>.*

in psql gives the necessary information.

To see the query that's used internally connect to a database with psql and supply an extra "-E" (or "--echo-hidden") option and then execute the above command.

After some searching, I was able to find the information_schema.routines table and the information_schema.parameters tables. Using those, one can construct a query for this purpose. LEFT JOIN, instead of JOIN, is necessary to retrieve functions without parameters.

SELECT routines.routine_name, parameters.data_type, parameters.ordinal_position
FROM information_schema.routines
    LEFT JOIN information_schema.parameters ON routines.specific_name=parameters.specific_name
WHERE routines.specific_schema='my_specified_schema_name'
ORDER BY routines.routine_name, parameters.ordinal_position;

If any one is interested here is what query is executed by psql on postgres 9.1:

SELECT n.nspname as "Schema",
  p.proname as "Name",
  pg_catalog.pg_get_function_result(p.oid) as "Result data type",
  pg_catalog.pg_get_function_arguments(p.oid) as "Argument data types",
 CASE
  WHEN p.proisagg THEN 'agg'
  WHEN p.proiswindow THEN 'window'
  WHEN p.prorettype = 'pg_catalog.trigger'::pg_catalog.regtype THEN 'trigger'
  ELSE 'normal'
 END as "Type"
FROM pg_catalog.pg_proc p
     LEFT JOIN pg_catalog.pg_namespace n ON n.oid = p.pronamespace
WHERE pg_catalog.pg_function_is_visible(p.oid)
      AND n.nspname <> 'pg_catalog'
      AND n.nspname <> 'information_schema'
ORDER BY 1, 2, 4;

You can get what psql runs for a backslash command by running psql with the -E flag.