1. Introduction
The Cauchy numbers [
1,
2,
3] of the first and second kind, respectively denoted by
and
, play important roles in many applications in number theory, combinatorics, and in different areas such as approximate integrals and difference-differential equations. These sequences of numbers are usually defined by means of the integral,
and:
where
is the rising factorial with
.
Bernoulli numbers are defined by the generating function:
One of the relations of the Bernoulli numbers with Stirling numbers of the second kind is:
where
represents the Stirling numbers of the second kind [
4] defined by:
In 1997, Kaneko [
5] defined certain variations of Bernoulli numbers in terms of the polylogarithm function:
which are called poly-Bernoulli numbers, denoted by
. More precisely, poly-Bernoulli numbers are defined by:
Parallel to this, certain variations of Cauchy numbers of the first kind were introduced by Komatsu [
2,
6], whose motivation was to relate the numbers to polylogarithm factorial functions:
These numbers are called poly-Cauchy numbers of the first kind, denoted by
. More precisely, these numbers are defined as follows:
The poly-Cauchy numbers possess several properties including an explicit formula:
where
represents the Stirling numbers of the first kind [
4] defined by:
the generating function:
and the following relation with Stirling numbers of the second kind:
Komatsu [
2,
6] also defined poly-Cauchy numbers of the second kind as follows:
Clearly,
. Similarly, these numbers possess the following properties: the explicit formula:
the generating function:
and a relation with Stirling numbers of the second kind:
Certain generalizations of poly-Cauchy numbers of the first and second kind were introduced by Cenkci and Young [
7]. This generalization was motivated by the concept of the Hurwitz–Lerch factorial zeta function defined by:
for
when
, Re
when
and
.
These numbers were called Hurwitz-type poly-Cauchy numbers of the first and second kind, denoted by
and
, which are respectively defined by:
and:
These numbers possess the following properties, which are parallel to those of poly-Cauchy numbers: the explicit formulas:
relations with Stirling numbers of the second kind:
and expressions of Hurwitz-type poly-Bernoulli numbers in terms of Hurwitz-type poly-Cauchy numbers:
Corcino et al. [
8,
9] extended the poly-Bernoulli numbers and defined the Hurwitz–Lerch-type multi-poly-Bernoulli numbers
as follows:
The numbers have the explicit formula:
Further generalization of poly-Cauchy numbers of the first and second kind was defined by Komatsu et al. [
10] using the multiple polylogarithm factorial function inspired by the following definition of multi-poly-Bernoulli numbers by Imatomi et al. [
11],
where:
is the multiple polylogarithm function. Komatsu et al. correspondingly defined a factorial version of the multiple polylogarithm factorial function as follows:
More precisely, the multi-poly-Cauchy numbers of the first and second kinds,
and
, are defined, respectively, as:
These numbers possess the following explicit formula:
In this paper, a multiple parameter Hurwitz–Lerch factorial zeta function will be introduced, which will be used to define certain multiple parameter Hurwitz–Lerch poly-Cauchy numbers. Several properties are established including generating functions, explicit formulas, and some relations involving the Stirling numbers.
2. Hurwitz–Lerch Multi-Poly-Cauchy Numbers
Combining the concepts of the Hurwitz–Lerch multi-poly-zeta function and multiple polylogarithm factorial function, we can define the Hurwitz–Lerch multi-factorial zeta function as follows:
Definition 1. The Hurwitz–Lerch multi-factorial zeta functionis defined by:wherewhenandwhenand.
Note that when
, Definition 1 gives:
the Hurwitz–Lerch factorial zeta function.
Comparing this with the left-hand side of (5) and (6), it would be logical to define the Hurwitz–Lerch multi-poly-Cauchy numbers of the first and second kinds as follows:
Definition 2. The Hurwitz–Lerch-type multi-poly-Cauchy numbers of the first kind are defined by: Definition 3. The Hurwitz–Lerch-type multi-poly-Cauchy numbers of the second kind are defined by: These numbers have the explicit formula involving Stirling numbers.
Theorem 1. For, we have: Proof. Now, working on the right-hand side, we have:
Comparing the coefficients completes the proof. ☐
The following theorem is an explicit formula for the Hurwitz–Lerch-type multi-poly-Cauchy numbers of the second kind, which can be shown similar to the proof of Theorem 1.
Theorem 2. For,
we have: Parallel to the results of Cencki and Young [
7], relations between Hurwitz–Lerch-type multi-poly-Cauchy numbers and Hurwitz–Lerch-type multi-poly-Bernoulli numbers can be shown using the orthogonality and inverse relations for Stirling numbers [
12]. By the orthogonality relations:
where
is a Kronecker symbol, it follows that:
Consequently, we obtain the following results:
Theorem 3. For the Hurwitz–Lerch-type multi-poly-Bernoulli numbers,
we have: Proof. Note that (3) can be expressed as:
Rewriting Equation (3) as:
for:
and:
the conclusion follows by applying Equation (10). ☐
Observe that when
the right-hand side of Equation (11) reduces to the single term
, so that:
which is [
7] (Theorem 2.5).
The next theorem contains result on Hurwitz–Lerch-type multi-poly-Cauchy numbers parallel to Theorem 3.
Theorem 4. For the Hurwitz–Lerch-type multi-poly-Cauchy numbers, we have:and: Proof. The proof can be shown parallel to Theorem 3. Now, (12) follows from (7) and (10) by considering:
and:
and (13) follows from (8) and (10) by considering:
and:
This completes the proof. ☐
Now, when
, the right-hand side of (12) and (13) reduces to the single term
and
, respectively, so that:
and:
which is [
7] (Theorem 2.6).
To obtain a kind of generalization of the results in [
7] (Theorem 2.7), we introduce modified Hurwitz–Lerch-type multi-poly-Bernoulli numbers, denoted by
, using the Hurwitz–Lerch multi-factorial zeta function as follows:
These numbers have the explicit formula involving Stirling numbers.
Lemma 1. For nonnegative integer n, we have: Proof. Working on the left-hand side of (15), we get:
Comparing the coefficients completes the proof. ☐
The next lemma is a result on the modified Hurwitz–Lerch-type multi-poly-Bernoulli numbers parallel to Theorem 3.
Lemma 2. For the Hurwitz–Lerch-type multi-poly-Bernoulli numbers, we have: Proof. Rewriting Lemma 1 as:
for:
and:
and the conclusion follows by applying (7). ☐
The next theorem contains the desired relationship between the Hurwitz–Lerch-type multi-poly-Cauchy numbers and the modified Hurwitz–Lerch-type multi-poly-Bernoulli numbers.
Theorem 5. For nonnegative integer n, we have: Proof. Using Lemma 1 and (12), we have:
Now, (17) can be found using Lemma 1 and (13). For (18), using (7) and Lemma 2 yields:
Similarly, (19) can be obtained using (8) and Lemma 2. ☐
When
, (16), (17), (18), and (19) yield:
where
is the modified Hurwitz–Lerch-type poly-Bernoulli numbers defined by: