We now note that is the Sheffer sequence for
Therefore,
(13)
is the Sheffer sequence for
So,
(14)
3.1 Explicit expressions
Recall that Barnes’ multiple Bernoulli polynomials are defined by the generating function as
(15)
where [6–8]. Let () with . The (signed) Stirling numbers of the first kind are defined by
Theorem 1
(16)
(17)
(18)
(19)
(20)
(21)
Proof Since
(22)
and
(23)
we have
So, we get (16).
Similarly, by
(24)
and (23), we have
Therefore, we get (19).
By (10) with (13), we get
Since
we obtain (17).
Similarly, by (10) with (14), we get
Since
we obtain (20).
Next, we obtain
Thus, we get the identity (18).
Similarly,
Thus, we get the identity (21). □
3.2 Sheffer identity
Theorem 2
(25)
(26)
Proof By (13) with
using (11), we have (25).
By (14) with
using (11), we have (26). □
3.3 Difference relations
Theorem 3
(27)
(28)
Proof By (9) with (13), we get
By (8), we have (27).
Similarly, by (9) with (14), we get
By (8), we have (28). □
3.4 Recurrence
Theorem 4
(29)
(30)
where is the nth ordinary Bernoulli number.
Proof By applying
(31)
[5], Corollary 3.7.2] with (13), we get
Now,
Since
is a series with order ≥1, by (17) we have
Since
(32)
we have
which is the identity (29).
Next, by applying (31) with (14), we get
Now,
By (20) we have
By (32), we have the identity (30). □
3.5 Differentiation
Theorem 5
(33)
(34)
Proof We shall use
(cf. [5], Theorem 2.3.12]). Since
with (13), we have
which is the identity (33). Similarly, with (14), we have the identity (34). □
3.6 More relations
The classical Cauchy numbers are defined by
(see e.g. [9, 10]).
Theorem 6
(35)
(36)
Proof For , we have
Observe that
Since
is a series with order ≥1, we have
Therefore, we obtain
which is the identity (35).
Next, for we have
Observe that
Thus, we have
Therefore, we obtain
which is the identity (36). □
3.7 Relations including the Stirling numbers of the first kind
Theorem 7 For , we have
(37)
(38)
Proof We shall compute
in two different ways. On the one hand,
On the other hand,
(39)
The second term of (39) is equal to
The first term of (39) is equal to
Therefore, we have, for ,
Thus, we get (37).
Next, we shall compute
in two different ways. On the one hand,
On the other hand,
(40)
The second term of (40) is equal to
The first term of (40) is equal to
From the proof of (36), we recall
Hence, the first term of (39) is equal to
Therefore, we get (38). □
3.8 Relations with the falling factorials
Theorem 8
(41)
(42)
Proof For (13) and (23), assume that . By (12), we have
Thus, we get the identity (41).
Similarly, for (13) and (23), assume that . By (12), we have
Thus, we get the identity (42). □
3.9 Relations 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, 12]).
Theorem 9
(43)
(44)
Proof For (13) and
(45)
assume that . By (12), similarly to the proof of (37), we have
Thus, we get the identity (43).
Next, for (14) and (45), assume that . By (12), similarly to the proof of (38), we have
Thus, we get the identity (44). □
3.10 Relations with higher-order Bernoulli polynomials
Bernoulli polynomials of order r are defined by
(see e.g. [5], Section 2.2]). In addition, Cauchy numbers of the first kind of order r are defined by
(see e.g. [13], (2.1)], [14], (6)]).
Theorem 10
(46)
(47)
Proof For (13) and
(48)
assume that . By (12), similarly to the proof of (37), we have
Thus, we get the identity (46).
Next, for (13) and (48), assume that . By (12), similarly to the proof of (38), we have
Thus, we get the identity (47). □