From (7) and (8), we note that
(16)
and
(17)
Let us assume that
(18)
From (11), (18) and , we note that
(19)
Now, we use the following fact:
(20)
For , by (11), (17) and (18), we get
(21)
Therefore, by comparing coefficients on both sides of (19) and (20), we obtain the following theorem.
Theorem 1 For , , , we have
We recall the following equation:
(22)
For , from (11), (17) and (18), we have
(23)
Therefore, by (19) and (23), we obtain the following theorem.
Theorem 2 For , , we have
Here we invoke the following identity:
(24)
Let us consider the following associated sequence:
(25)
For , by (19) and (25), we get
(26)
Let us assume that
(27)
For , by (11), (27) and , we get
(28)
For , from (11), (25) and (27), we can derive the following equation:
(29)
Therefore, by (26) and (29), we obtain the following theorem.
Theorem 3 For , , we have
Here we use the following identity:
(30)
Let us consider the following associated sequence:
(31)
For , from (19) and (31), we have
(32)
Let us assume that
(33)
Then, from (28) and (33), we note that, for ,
(34)
For , by (11), (32) and (33), we get
(35)
Therefore, by (32) and (35), we obtain the following theorem.
Theorem 4 For , , we have
Now, we utilize the following identity:
(36)
Let us consider the following associated sequence:
(37)
For , from (19) and (37), we have
(38)
Let us assume that
(39)
We observe that
(40)
From (11), (39), (40) and , we can derive the following equation:
(41)
For , by (11), (37), (39) and (41), we get
(42)
Therefore, by (38) and (42), we obtain the following theorem.
Theorem 5 For , , we have
Now, we recall the following identity:
(43)
Let us consider the following associated sequence:
(44)
For , from (19) and (44), we can derive the following equation:
(45)
Let us assume that
(46)
We observe that
(47)
For , by (11), (46), (47) and , we get
(48)
For , from (11), (44), (46) and (48), we have
(49)
Therefore, by (45) and (49), we obtain the following theorem.
Theorem 6 For , , we have
Here we invoke the following identity:
(50)
Let us consider the following associated sequence:
(51)
From (19) and (51), we note that
(52)
Let us assume that
(53)
For , from (48) and (49), we have
(54)
For , from (11), (51), (53) and (50), we can derive the following identity:
(55)
Therefore, by (52) and (55), we obtain the following theorem.
Theorem 7 For , , we have
Here we use the following identity:
(56)
Let us consider the following associated sequence:
(57)
By (19) and (57), we get
(58)
Let us assume that
(59)
We see that
(60)
For , from (11), (59), (60) and , we have
(61)
From (61), by the same method of (48), we get
(62)
For , by (11), (56), (57), (59) and (62), we get
(63)
By the same method, we can derive the following identity from (63):
(64)
By comparing coefficients on both sides of (58) and (64), we get
(65)
Remark Recently, several authors have studied the q-extension of harmonic and hyperharmonic numbers (see [11–13]).