1. Introduction and Motivation
Recently, Choi et al. ([
1], Definition 1) defined a new number sequence, which we restate as follows. For two indeterminates,
and
, the sequences
are defined by
and
when
.
Here and throughout the paper, we shall make use of the following notations for shifted factorials. For an indeterminate
x and a non-negative integer
n, they are defined by
and
The
r-Whitney–Lah numbers
are defined as [
2,
3,
4]:
where
and
are the generalized rising and falling factorials with
. Comparing (
1) and (
2), when
, we obtain
.
Choi et al. ([
1], Lemma 1) derived the explicit expression of
:
and a summation formula, which is equivalent to
where
and
denote the signed and unsigned Stirling numbers of the first kind defined by the generating functions [
5,
6,
7]:
Obviously, when
, we have the relation
where
is the Lah numbers defined by the generating function [
5,
6]:
For function
, denote its difference by
and
n-order difference
where the symbol
is called the difference operator. By means of induction, it is not difficult to obtain
and
Let
denote the coefficient of
in the formal power series
. Then, we have the effective lemma [
6,
7] below, which will be frequently used in the next sections to establish summation formulas.
Lemma 1. For the double indexed sequence , subject to when , and sequence that have generating functionswhere , then we have In this paper, we shall continue to explore the properties and satisfied identities of the numbers
that are not listed by Choi et al. in [
1]. In addition, the
r-Whitney–Lah numbers will be used as special cases to show some of our results. This study is an extension of the study of generalized Lah numbers and is instructive for the further study of other combinatorial sequences such as Stirling numbers, harmonic numbers, Cauchy numbers, Catalan numbers, etc.
The rest of this paper is organized as follows. In the next section, we shall use a difference operator to regain the explicit expression of the number
and some of its other characteristics, such as its recurrence relations, generating function and inversion formula, as well as some summation formulas involving Lah numbers and Stirling numbers of the first kind. In
Section 3, some formulas, concerning classical and generalized harmonic numbers, will be established via the sequence
and Lemma 1. In
Section 4, we shall derive several summation formulas concerning Cauchy numbers. Then, the paper will end in
Section 5 with comments and some summation formulas involving Catalan numbers.
2. Some Results Involving , Lah and Stirling Numbers
In this section, firstly, we shall derive a trivial recursive relation for
by using (
1) and its explicit expression by a difference operator, as well as a nontrivial recursive relation by a linear relation. Then, we shall establish the generating function of
and some transformation formulas.
Proposition 1 (
Trivial recursive relation)
. For and , the following recursive relation holds Proof. By using the (
1), we have
The left hand side can be rewritten as
Then, the proof follows by comparing the coefficients of
. □
Proposition 2 (
Explicit expression)
. For and , we have the explicit expression Proof. By utilizing (
3) and (
4), we can evaluate the
k-order difference
Letting
and noting that
for
, we obtain
which leads us to the explicit expression of
. □
Proposition 3 (
Nontrivial recursive relation)
. For and , the following recursive relation holds Proof. By means of the linear relation
we have the following equation:
□
By specifying the parameter in Proposition 3, we obtain the following recursive relation on Lah numbers:
Corollary 1. For , the following recursive relation holds: Proposition 4 (
Exponential generating function)
. Proof. According to the explicit expression of
(Proposition 2), we have
Then the proof follows by using the fact
□
Proposition 5. The ordinary generating function of satisfies the difference-differential equation Proof. By means of the recursive relation Proposition 1, we have
Under the replacement
, we get the equation
which confirms the result stated in the proposition. □
Proposition 6. For and , the following formula holds: Proof. Using the explicit expression of
(Proposition 2), we have
□
For the particular case of , the above Proposition 6 reduces to the corollary below.
Corollary 2. For , the following formula holds: Similar to the proof of Proposition 6, we can get the following alternating summation formula:
Proposition 7 (
Orthogonality relation)
. For and , the following formula holds:where be the Kronecker symbol By letting , we can obtain, from the above proposition, the following known formula on Lah numbers.
Corollary 3. For , the following formula holds: Proposition 8 (
Reversion formula)
. For two sequences and , the following equivalent relations hold: Proof. If the two sequences
and
satisfy
then we can evaluate the sum
By means of Proposition 7, the inner sum in the last line can be rewritten as
We therefore obtain the following formula:
And vice versa. □
Proposition 9. For and , the following formula holds: Proof. By means of Lemma 1 and the exponential generating functions of
and
, we can manipulate the sum
Then, the proof follows by evaluating the coefficients
□
Letting in Proposition 9, we have the following summation formula involving Lah numbers and Stirling numbers of the first kind:
Corollary 4. For , the following formula holds: Letting
in Proposition 9, we have the following summation formula involving
r-Whitney–Lah numbers and Stirling numbers of the first kind: For
, the following formula holds:
Alternatively, by letting in Proposition 9, we have another formula below, involving signed and unsigned Stirling numbers of the first kind.
Corollary 5. For , the following formula holds: Proof. By setting
, the formula stated in Proposition 9 can be rewritten as
Then, the identity desired follows by employing the known Formula ([
8], Eq. 6.16)
□
Proposition 10. For and , the following formula holds: Proof. In ([
1], Lemma 1), Choi et al. obtained the following transformation formula:
Combining it with Proposition 9, we can obtain the desired symmetric transformation formula. □
For the particular case of , the Proposition 10 reduces the following identity.
Corollary 6. For , the following formula holds: In fact, we have the closed expression
which are recorded in ([
8], Eqs. 6.16 and 6.21) by Graham et al.
3. Summation Formulas concerning Harmonic Numbers
It is well known that the classical harmonic numbers are defined by
and its generating function is given by
Harmonic numbers have wide applications in number theory, combinatorics, and computer science. Their properties and identities have been explored extensively. In addition, many authors also have studied other harmonic-like numbers defined in various ways. For instance, Cheon and El-Mikkawy [
9,
10] (also see [
7,
11]) studied the following multiple harmonic-like numbers, which reduce, when
, to the ordinary harmonic numbers
and obtained its generating function:
In [
7], Guo and Chu also studied the alternating harmonic numbers
as well as the multiple alternating harmonic numbers
and obtained their generating functions:
Now, we further explore the summation formulae concerning (generalized) harmonic numbers and the sequence , whose particular cases reduces to several interesting identities on Stirling numbers.
Proposition 11. For and , the following formula holds: Proof. With the aid of Lemma 1 and the generating functions of
and
, the sum on the left hand side can be evaluated as
Extracting the coefficients of
from the last line
we therefore confirm the desired formula. □
For the particular cases
and
, Proposition 11 reduces the following well-known formulas
Analogously, we have the following formula on alternating harmonic numbers .
Proposition 12. For and , the following formula holds: Particularly, when and , we have, respectively, the formula below.
Corollary 7. For , the following two formulas hold: When
in Propositions 11 and 12, we have, respectively,
Proposition 13 (
Explicit expression of )
. Proof. According to the proof of Proposition 9 and the generating function of
, when
, we have
Combining this with (5), we obtain the explicit expression of
. □
This explicit expression was also found by Cheon and El-Mikkawy [
10], but there does not exist such an elegant expression for
as that which Guo and Chu pointed out in [
7].
Proposition 14. For and , the following formula holds: Proof. By employing Lemma 1 and the generating functions of
and
, we have
By extracting the coefficients
we therefore obtain the formula stated in the proposition. □
Setting
in Proposition 14, we obtain that
Letting and in Proposition 14, we obtain the following summation formulas.
Corollary 8. For , the following two formulas hold: Analogously, for the multiple alternating harmonic numbers
, we have the expression
The special cases
and
of (
5) lead to the following identities.
Corollary 9. For , the following two formulas hold: Proposition 15. For and , the following two formulas hold: Proof. Letting
in Proposition 4, we obtain the generating function
By means of Lemma 1 and the generating function of
, we can evaluate
By using the generating function of
and the coefficient
we can evaluate the coefficients
which confirms the identity (
6). The second identity (
7) follows by using Proposition 9. □
Proposition 16. For , the following relation holds: Proof. By setting
, the Equation (
8) becomes
Keeping in mind Corollary 4, we have the equation
which completes the proof. □
By telescoping, we can obtain, from Proposition 16, the following summation formula.
Corollary 10. For , the following relation holds: Proposition 17. For , the following relation holds: Proof. By letting
in (
8), we obtain the identity
According to Corollary 5, we have the summation formula
Combining the above two equations, we can complete the proof. □
Proposition 18. For and , the following formula holds:where denotes the generalized harmonic numbers defined by Proof. It is easy to verify the relation
By differentiating, with respect to
, both sides of Proposition 9, we can obtain the desired identity. □
For the special case of in Proposition 18, we can find the formula below.
Corollary 11. For , the following formula holds: By denoting , we have the following proposition.
Proposition 19. For and , the following formula holds: Proof. In ([
12], Eq. 20), Chen obtained the generating function
By employing Lemma 1, we can compute the sum
Evaluating the coefficient
we therefore obtain the formula stated in the proposition. □
Let
in Proposition 19, we have that
For the special case of , Proposition 19 gives the formula below.
Corollary 12. For , the following formula holds 4. Formulas concerning and Cauchy Numbers
The first- and second-kind Cauchy numbers
and
are defined, respectively, by the integrals (cf. [
5,
13])
Their generating functions are given by
They satisfy the following relations:
where the latter one corrects the error recorded in ([
5], p. 294).
Similar to the last section, by means of Lemma 1, we can establish the following summation formulas.
Proposition 20. For and , the following formulas hold: Proof. For the first Formula (
9), by applying Lemma 1 to the left hand side, we can evaluate
The proof follows by extracting the coefficient
Analogously, we can obtain the second one (
10). □
By setting
in Proposition 20, we obtain
When setting
, Proposition 20 reduces to the following two formulas concerning Lah numbers, which can be found in ([
6], Eqs. 3.1 and 3.2) (also see [
13]).
Alternatively, by letting
in Proposition 20, we obtain another two summation formulas.
Corollary 13. For , the following formulas hold: Further, it is not difficult to confirm ([
1], Eq. 35)
Then, by differentiating, with respect to
, both sides of (
9) and (
10), we obtain the formulas below.
Proposition 21. For and , the following formulas hold: Particularly, when , the above proposition reduce to the following identities.
Corollary 14. For , the following formulas hold: 5. Concluding Comments
By means of the numbers
and Lemma 1, we may establish more summation formulas. For example, recalling the Catalan numbers
[
14] defined by
with the generating function
we can obtain the following summation formula:
By specifying
,
and
, the above formula reduces to the identities below.
By differentiating, with respect to
, both sides of (
11), we obtain, for
, the following formula involving harmonic numbers.
For
, it reduces to the following summation formula:
In this paper, we mainly obtain some properties of the numbers and some identities concerning them and other combinatorial numbers, which is different from the results obtained by Choi et al., which mainly obtain some transformation formulas regarding hypergeometric series through .