From (2), (3), and (19), we note that
(28)
and
(29)
It is not difficult to show that
(30)
and
(31)
From (14), we have
(32)
Therefore, by (32), we obtain the following theorem.
Theorem 1 For , we have
Alternatively,
(33)
Therefore, by (33), we obtain the following theorem.
Theorem 2 For , let . Then we have
Remark By the same method, we get
(34)
and
(35)
From (20) and (28), we have
(36)
From (31), we note that
(37)
Therefore, by (36) and (37), we obtain the following theorem.
Theorem 3 For , we have
Remark By the same method as Theorem 3, we get
(38)
From (28), we note that
(39)
and
By (24), (39), and (40), we get
(41)
Thus, by (41), we see that
(42)
Therefore, by (42), we obtain the following theorem.
Theorem 4 For , we have
Remark By the same method as Theorem 4, we get
(43)
From (12), we note that
(44)
Thus, by (44), we see that
(45)
and
(46)
From (45) and (46), we have
(47)
Thus, by (47), we get
(48)
Therefore, by (48), we obtain the following theorem.
Theorem 5 For , we have
It is easy to see that
(49)
and
(50)
By the same method as Theorem 5, we get
(51)
From (20) and (28), we have
(52)
Now, we observe that
(53)
Therefore, by (52) and (53), we obtain the following theorem.
Theorem 6 For , we have
Remark By the same method as Theorem 6, we get
(54)
From (21), we have
(55)
and
(56)
By (22) and (28), we get
(57)
and
(58)
Therefore, by (57) and (58), we obtain the following theorem.
Theorem 7 For , we have
Remark By the same method as Theorem 7, we get
(59)
From (22), (28), and (29), we have
(60)
and
(61)
By (14) and (27), we get
(62)
Now, we observe that
(63)
Therefore, by (62) and (63), we obtain the following theorem.
Theorem 8 For , we have
where .
Remark By the same method as Theorem 8, we get
(64)
By (23), we get
(65)
By the same method as (65), we get
(66)
Now, we compute the following equation in two different ways:
On the one hand,
(67)
On the other hand,
(68)
Therefore, by (67) and (68), we obtain the following theorem.
Theorem 9 For with , let . Then we have
Remark By the same method as Theorem 9, we get
where .
Let us consider the following two Sheffer sequences:
(69)
and
(70)
Let
(71)
Then, by (26), we get
(72)
Therefore, by (71) and (72), we obtain the following theorem.
Theorem 10 For , we have
Remark By the same method as Theorem 10, we have
(73)
For , , , with , let us assume that
(74)
From (26), we have
(75)
Therefore, by (75) and (76), we obtain the following theorem.
Theorem 11 For with , , we have
Remark By the same method as Theorem 11, we get
For and , let us assume that
(76)
By (26), we get
(77)
Therefore, by (77) and (78), we obtain the following theorem.
Theorem 12 For , we have
Remark By the same method as Theorem 12, we get
(78)