Let us take . Then can be expressed as a ℚ-linear combination of as follows:
(10)
Now, let us define the operator by
(11)
Then, by (10) and (11), we set
(12)
From (1), we note that
(13)
By (2), (3) and (13), we get
(14)
From (12) and (14), we get
(15)
For , let us take the r th derivative of in (15) as follows:
(16)
Thus, by (16), we get
(17)
From (11) and (17), we have
(18)
where .
Therefore, by (10) and (18), we obtain the following theorem.
Theorem 1 For , let with .
Then we have
Let us assume that . Then by Theorem 1, we get
(19)
where
(20)
From (6) and (20), we have
(21)
By (19) and (21), we get
(22)
Therefore, by (22), we obtain the following theorem.
Theorem 2 For , we have
In particular, if we take , then we have
(23)
where
(24)
By (8) and (24), we get
(25)
From (23) and (25), we have
Let us take with
(26)
Then we have
Continuing this process, we get
(27)
From (27), we have
(28)
By (6), we get
(29)
From (28) and (29), we have
(30)
By Theorem 1, can be expressed by
(31)
where
(32)
Thus, by (31) and (32), we get
(33)
Therefore, by (31) and (33), we obtain the following theorem.
Theorem 3 For , we have