From definition (1), is the Sheffer sequence for the pair
So,
(12)
3.1 Explicit expressions
Recall that Barnes’ multiple Bernoulli polynomials are defined by the generating function as follows:
(13)
where [8, 9]. Let () with . The (signed) Stirling numbers of the first kind are defined by
Theorem 1
(14)
(15)
(16)
(17)
Proof Since
(18)
and
(19)
we have
So, we get (14).
By (9) with (12), we get
Thus, we obtain
which is identity (15).
Next,
Thus, we obtain (16).
Finally, we obtain that
Thus, we get identity (17). □
3.2 Sheffer identity
Theorem 2
(20)
Proof By (12) with
using (10), we have (20). □
3.3 Difference relations
Theorem 3
(21)
Proof By (8) with (12), we get
By (7), we have (21). □
3.4 Recurrence
Theorem 4
(22)
where is the nth ordinary Bernoulli number.
Proof By applying
(23)
[7, Corollary 3.7.2] with (12), we get
Now,
Observe that
is a series with order ≥1. Since
we have
(24)
Since
the first term in (24) is
Since
(25)
the second term in (24) is
The third term in (24) is
Thus we have identity (22). □
3.5 Differentiation
Theorem 5
(26)
Proof We shall use
(cf. [7, Theorem 2.3.12]). Since
with (12), we have
which is identity (26). □
3.6 More relations
The classical Cauchy numbers are defined by
(see e.g. [1, 10]).
Theorem 6
(27)
Proof For , we have
The third term is
By (25), the second term is
Observe that
with
a series with order ≥1.
Now, the first term is
Altogether, we obtain
from which identity (27) follows. □
3.7 A relation including the Stirling numbers of the first kind
Theorem 7 For , we have
(28)
Proof We shall compute
in two different ways. On the one hand, it is
On the other hand, it is
(29)
The third term of (29) is equal to
The second term of (29) is equal to
The first term of (29) is equal to
Therefore, we get, for ,
Dividing both sides by , we get (28). □
3.8 A relation with the falling factorials
Theorem 8
(30)
Proof For (12) and (19), assume that . By (11), we have
Thus, we get identity (30). □
3.9 A relation with higher-order Frobenius-Euler polynomials
For with , the Frobenius-Euler polynomials of order r, are defined by the generating function
(see e.g. [11]).
Theorem 9
(31)
Proof For (12) and
(32)
assume that . By (11), similarly to the proof of (28), we have
Thus, we get identity (31). □
3.10 A relation with higher-order Bernoulli polynomials
Bernoulli polynomials of order r are defined by
(see e.g. [7, Section 2.2]). In addition, the Cauchy numbers of the first kind of order r are defined by
(see e.g. [[12], (2.1)], [[13], (6)]).
Theorem 10
(33)
Proof For (12) and
(34)
assume that . By (11), similarly to the proof of (28), we have
Thus, we get identity (33). □