We now note that , and are the Appell sequences for
So,
(14)
(15)
(16)
In particular, we have
(17)
(18)
(19)
Notice that
3.1 Explicit expressions
Write . Let () with .
Theorem 1
(20)
(21)
(22)
(23)
(24)
Proof By (1), (2) and (3), we have
So, we get (20).
We also have
Thus, we get (21).
In [5] we obtained that
So,
which is identity (22).
In [5] we obtained that
where are the Stirling numbers of the second kind, defined by
Thus,
which is identity (23).
By (11) with (16), we have
Thus, we get (24). □
3.2 Sheffer identity
Theorem 2
(25)
Proof By (16) with
using (12), we have (25). □
3.3 Recurrence
Theorem 3
(26)
where is the nth ordinary Bernoulli number.
Proof By applying
[[4], Corollary 3.7.2] with (16), we get
Now,
Since
we have
Since
is a delta series, we get
Therefore, by
we obtain
which is identity (26). □
3.4 A more relation
Theorem 4
(27)
Proof For , we have
Observe that
Thus,
Since
and the fact that
is a delta series, we have
Therefore, we obtain the desired result. □
Remark After n is replaced by , identity (27) becomes the recurrence formula (26).
3.5 Relations with poly-Bernoulli numbers and Barnes’ multiple Bernoulli numbers
Theorem 5
(28)
Proof We shall compute
in two different ways. On the one hand,
On the other hand,
Here, . Thus, we get (28). □
3.6 Relations with the Stirling numbers of the second kind and the falling factorials
Theorem 6
(29)
Proof For (16) and , assume that
By (13), we have
Thus, we get identity (29). □
3.7 Relations with the Stirling numbers of the second kind and the rising factorials
Theorem 7
(30)
Proof For (16) and , assume that . By (13), we have
Thus, we get identity (30). □
3.8 Relations with higher-order Frobenius-Euler polynomials
Theorem 8
(31)
Proof For (16) and
assume that . By (13), we have
Thus, we get identity (31). □
3.9 Relations with higher-order Bernoulli polynomials
Bernoulli polynomials of order r are defined by
(see, e.g., [[4], Section 2.2]).
Theorem 9
(32)
Proof For (16) and
assume that . By (13), we have
Thus, we get identity (32). □