1. Introduction
Special polynomials (like Bernoulli, Euler, Hermite, Laguerre, etc.) have great importance in applied mathematics, mathematical physics, quantum mechanics, engineering, and other fields of mathematics. Particularly the family of special polynomials is one of the most useful, widespread, and applicable families of special functions. Recently, the aforementioned polynomials and their diverse extensions have been studied and introduced in [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14].
In this paper, the usual notations refer to the set of all complex numbers
, the set of real numbers
, the set of all integers
, the set of all natural numbers
, and the set of all non-negative integers
, respectively. The classical Bernoulli polynomials
are defined by
Upon setting
in (
1), the Bernoulli polynomials reduce to the Bernoulli numbers, namely,
. The Bernoulli numbers and polynomials have a long history, which arise from Bernoulli calculations of power sums in 1713 (see [
9]), that is
The Bernoulli polynomials have many applications in modern number theory, such as modular forms and Iwasawa theory [
11].
In 1924, Nörlund [
13] introduced the Bernoulli polynomials and numbers of order
For
, and
, Su and Komatsu [
10] defined the hypergeometric Bernoulli polynomials
of order
by means of the following generating function:
where
is called the confluent hypergeometric function (see [
14]) with
for
and
. When
,
are the higher-order generalized hypergeometric Bernoulli numbers. When
, the higher-order hypergeometric Bernoulli polynomials
, which are studied by Hu and Kim in [
9]. When
, we have that
are the hypergeometric Bernoulli polynomials which are defined by Howard [
7,
8] as
For
in (
3), we have
.
The Lagrange polynomials in several variables, which are known as the Chan–Chyan–Srivastava polynomials [
2], are defined by means of the following generating function:
and are represented by
Altin and Erkus [
1] introduced the multivariable Lagrange–Hermite polynomials given by
where
In the special case when
in (
7), the polynomials
reduce to the familiar (two-variable) Lagrange–Hermite polynomials considered by Dattoli et al. [
3]:
The multivariable (Erkus–Srivastava) polynomials
are defined by the following generating function [
6]:
which are a unification (and generalization) of several known families of multivariable polynomials including the Chan–Chyan–Srivastava polynomials
in (
5) and multivariable Lagrange–Hermite polynomials (
7).
By (
9), the Erkus–Srivastava polynomials
satisfy the following explicit representation (cf. [
6]):
which is the generalization of Relation (
6).
In this paper, we introduce the multivariable unified Lagrange–Hermite-based hypergeometric Bernoulli polynomials and investigate some of their properties. Then, we derive multifarious connected formulas involving the Miller–Lee polynomials, the Laguerre polynomials polynomials, the Lagrange Hermite–Miller–Lee polynomials.
2. Lagrange–Hermite-Based Hypergeometric Bernoulli Polynomials
By means of (
3) and (
9), we consider a unification of the hypergeometric Bernoulli polynomials
of order
and the multivariable (Erkus–Srivastava) polynomials
. Thus, we define the multivariable unified Lagrange–Hermite-based hypergeometric Bernoulli polynomials
of order
by means of the following generating function:
where
,
for
and
. Upon setting
, we have
, which we call the multivariable Lagrange–Hermite-based hypergeometric Bernoulli polynomials of order
where
for
and
. Furthermore, note that
Remark 1. In the case and , we get , which we call the Lagrange–Hermite-based hypergeometric Bernoulli polynomials of order α: Remark 2. When and we acquire , which we call the Lagrange-based hypergeometric Bernoulli polynomials of order α, and which are defined by When
in (
14), we have
, which we call the Lagrange-based hypergeometric Bernoulli numbers of order
.
We now investigate some properties of .
Theorem 1. The following summation formula:holds for . Proof. By (
11), we have
which gives the asserted Formula (
15). □
Theorem 2. The following summation formula:holds for.
Proof. By using (
13), we have
which gives the asserted result (
16). □
We give the following theorem:
Theorem 3. The following summation formula:holds for.
Proof. Using definition (
11), we have
which provides the claimed result (
17). □
We state the following theorem:
Theorem 4. The following summation formulas for the higher-order generalized hypergeometric Lagrange–Hermite–Bernoulli polynomials hold:and Proof. For
and
in (
13), we have
Therefore, by integrating (
20) with weight
, we obtain
which completes the proof. □
Theorem 5. The following summation formula for the higher-order generalized hypergeometric Lagrange–Hermite–Bernoulli polynomials holds: Proof. For
and
in (
13), we have
Comparing the coefficients of
in both sides, we get the result (
21). □
We give the following derivative property:
Theorem 6. The following derivative property for the higher-order hypergeometric generalized Lagrange–Hermite–Bernoulli polynomials holds: Proof. Start with
which implies the asserted result (
22). □
Theorem 7. The following summation formula involving the higher-order generalized hypergeometric Lagrange–Hermite–Bernoulli polynomials and higher-order generalized hypergeometric Lagrange–Bernoulli polynomials holds true: Proof. The proof is similar to Theorem 3. □
3. Some Connected Formulas
The generation functions (
13) and (
14) can be exploited in a number of ways and provide a useful tool to frame known and new generating functions in the following way:
As a first example, we set
,
,
in (
13) to get
where
are called the Miller–Lee polynomials (see [
4]).
Another example is the definition of higher-order hypergeometric Bernoulli–Hermite–Miller–Lee polynomials
given by the following generating function:
which for
reduces to
where
are called the Lagrange Hermite–Miller–Lee polynomials.
Putting
into (
25) gives
where
are called the higher-order hypergeometric Bernoulli–Miller–Lee polynomials.
We now give some connected formulas as follows:
Theorem 8. The following implicit summation formula involving higher-order hypergeometric Lagrange–Hermite–Bernoulli polynomials , Bernoulli–Miller–Lee polynomials and Miller–Lee polynomials holds: Proof. For
and
in (
13) and using (
27), we have
which by using binomial expansion takes the form
which implies the asserted result (
28). □
Theorem 9. The following implicit summation formula involving higher-order Lagrange–Hermite–Bernoulli polynomials and Miller–Lee polynomials holds: Proof. On replacing
x with
and
with
, respectively, in (
13), we have
which yields the claimed result (
29). □
Theorem 10. The following implicit summation formula involving higher-order Lagrange–Hermite–Bernoulli polynomials and Miller–Lee polynomials holds: Proof. For
and
in (
13), we have
Multiplying both the sides by
, we have
Now, replacing
n by
in the above equation, we get
Comparing the coefficient of
, we get the result (
30). □
Now, we shall focus on the connection between the higher-order generalized hypergeometric Lagrange–Hermite–Bernoulli polynomials and Laguerre polynomials .
For
,
,
and
in Equation (
11), we have
where
are called generalized higher-order hypergeometric Bernoulli–Laguerre polynomials.
When
in (
31),
reduces to ordinary Laguerre polynomials
(see [
14]).
Theorem 11. The following implicit summation formula involving higher-order Lagrange–Hermite–Bernoulli polynomials and Laguerre polynomials holds: Proof. By replacing
x with
and setting
,
in (
13), we have
Multiplying both sides
, we have
which gives
which yields the asserted result (
32). □
Theorem 12. The following implicit summation formula involving higher-order hypergeometric Lagrange–Hermite–Bernoulli polynomials and Laguerre polynomials holds true: Proof. By replacing
x with
and setting
,
, and
in Equation (
11), we have
which yields the asserted result (
33). □
Theorem 13. The following implicit summation formula involving the Lagrange–Hermite–Bernoulli polynomials and Laguerre polynomials holds true: Proof. Replacing
with
and
in (
13), we have
which implies the claimed result (
34). □
Theorem 14. The following implicit summation formula involving higher-order Lagrange–Hermite–Bernoulli polynomials and Laguerre polynomials holds: Proof. For
,
,
and replacing
x with
in (
13), we have
which gives the claimed result (
35). □
Theorem 15. The following implicit summation formula involving higher-order Lagrange–Hermite–Bernoulli polynomials and generalized Laguerre-Bernoulli polynomials holds: Proof. By (
13), we write
which yields the asserted result (
36). □