Does vertica support derivative column concept in case statement.

What version of Vertica are you running?

It's relatively easy to accomplish your goal using the LISTAGG function which is available in Vertica 9.1.1-4 and above.

Example:

dbadmin=> SELECT * FROM tab1;
 id | rank | columnname
----+------+------------
  1 |    1 | A
  1 |    2 | B
  1 |    3 | C
  1 |    4 | D
(4 rows)

dbadmin=> SELECT a.id, a.rank, a.columnname, listagg(b.columnname)
dbadmin->  FROM tab1 a
dbadmin->  JOIN tab1 b
dbadmin->    ON a.id = b.id
dbadmin->   AND b.rank <= a.rank
dbadmin->  GROUP BY a.id, a.rank, a.columnname;
 id | rank | columnname | listagg
----+------+------------+---------
  1 |    1 | A          | A
  1 |    2 | B          | A,B
  1 |    3 | C          | A,B,C
  1 |    4 | D          | A,B,C,D
(4 rows)

See:
https://www.vertica.com/docs/9.2.x/HTML/Content/Authoring/SQLReferenceManual/Functions/Aggregate/LISTAGG.htm

Another option is to use the LAG analytic function:

dbadmin=> SELECT * FROM tab1;
 id | rank | columnname
----+------+------------
  1 |    1 | A
  1 |    2 | B
  1 |    3 | C
  1 |    4 | D
  2 |    1 | X
  2 |    2 | Y
  2 |    3 | Z
(7 rows)

dbadmin=> SELECT id, rank, columnname,
dbadmin->        CASE
dbadmin->        WHEN rank = 1
dbadmin->          THEN columnName
dbadmin->        WHEN rank = 2
dbadmin->          THEN LAG(columnName) OVER (PARTITION BY id ORDER BY rank) || ',' || columnName
dbadmin->        WHEN rank = 3
dbadmin->          THEN LAG(columnName, 2) OVER (PARTITION BY id ORDER BY rank) || ',' ||
dbadmin->            LAG(columnName) OVER (PARTITION BY id ORDER BY rank) || ',' || columnName
dbadmin->        WHEN rank = 4
dbadmin->          THEN LAG(columnName, 3) OVER (PARTITION BY id ORDER BY rank) || ',' ||
dbadmin->            LAG(columnName, 2) OVER (PARTITION BY id ORDER BY rank) || ',' || LAG(columnName) OVER (PARTITION BY id ORDER BY rank) || ',' || columnName
dbadmin->        END as aggregate
dbadmin->   FROM tab1
dbadmin->   ORDER BY id, rank;
 id | rank | columnname | aggregate
----+------+------------+-----------
  1 |    1 | A          | A
  1 |    2 | B          | A,B
  1 |    3 | C          | A,B,C
  1 |    4 | D          | A,B,C,D
  2 |    1 | X          | X
  2 |    2 | Y          | X,Y
  2 |    3 | Z          | X,Y,Z
(7 rows)