In this section, we use umbral calculus to have alternative ways of obtaining our results. Let us consider the following Sheffer sequences:
(2.1)
Then, by (1.13), we assume that
(2.2)
From (1.14) and (2.2), we have
(2.3)
Therefore, by (2.2) and (2.3), we obtain the following theorem.
Theorem 2.1 For , we have
Let us consider the following two Sheffer sequences:
(2.4)
Let us assume that
(2.5)
By (1.14) and (2.4), we get
(2.6)
Therefore, by (2.5) and (2.6), we obtain the following theorem.
Theorem 2.2 For , we have
Consider
(2.7)
Let us assume that
(2.8)
By (1.14), we get
(2.9)
Therefore, by (2.8) and (2.9), we obtain the following theorem.
Theorem 2.3 For , we have
Let us assume that
(2.10)
From (1.14), (2.1), and (2.10), we have
(2.11)
Therefore, (2.10) and (2.11), we obtain the following theorem.
Theorem 2.4 For , we have
Note that . Thus, we have
(2.12)
From (2.12), we have
(2.13)
By (2.11) and (2.13), we also see that
(2.14)
Therefore, by (2.10) and (2.14), we obtain the following theorem.
Theorem 2.5 For , we have
Let us assume that
(2.15)
From (1.14), (2.4), and (2.15), we have
(2.16)
From (2.13) and (2.16), we have
(2.17)
For , by (1.5) and (2.17), we get
(2.18)
Therefore, by (2.15) and (2.18), we obtain the following theorem.
Theorem 2.6 For , we have
Let us assume that . For , by (2.18), we get
(2.19)
For , by (2.17), we get
(2.20)
Therefore, by (2.15), (2.19), and (2.20), we obtain the following theorem.
Theorem 2.7 For , we have
Let us assume that
(2.21)
Then, by (1.14), (2.7), and (2.21), we get
(2.22)
By (2.13) and (2.22), we get
(2.23)
Therefore, by (2.21) and (2.23), we obtain the following theorem.
Theorem 2.8 For , we have
Remark From (2.22), we have
(2.24)
Thus, by (2.21) and (2.24), we get