1. Introduction
Appell polynomials are one of the largest class of polynomials. The classical Appell polynomials comprise the Bernoulli, Euler, Appell-Hermite, generalized Bernoulli, and generalized Euler polynomials, as well as other examples in [
1,
2,
3,
4].
The sequence of Appell polynomials is defined by either of the following identities:
where denotes the Appell numbers.
According to the alternative definition in [
3], the sequence of Appell polynomials
for
can be equivalently defined by
where the generating function satisfies
, and thus (
1) is identical to (ii).
Appell polynomials appear in various problems in pure and applied mathematics related to differential equations, approximation theory, interpolation, and summation methods. More examples may be found in [
1,
2,
5,
6] (and references therein). Moreover, in theoretical applications, degenerate versions of Appell polynomials
of a complex variable
have been extensively studied [
7,
8,
9,
10,
11]. These polynomials are defined by the generating functions
where
and
are obtained from the degenerate exponential functions. That is, the exponential functions
and
are replaced by
and
, respectively:
Here, it is noted that
and
, because
as
. Since Carlitz in [
12,
13] introduced degenerate formulas for special numbers and polynomials, the degenerate complex Appell polynomials have been extensively studied to find useful properties and identities. Specifically, degenerate Bernoulli and Euler polynomials of complex variables were introduced in [
12], whereas degenerate gamma functions and Laplace transforms were introduced in [
14], and their modifications in [
9]. The study of degenerate versions of known special numbers and polynomials provides several useful identities and related properties. Although the original sequence
satisfies conditions (i) and (ii), the sequence
of degenerate polynomials no longer satisfies these conditions.
To retain the crucial properties of the Appell sequence, we introduce a new type of degenerate Appell polynomials, which are obtained by partially degenerate generating functions. Some examples are provided in
Table 1. The aim of this paper is to present degenerate complex Appell polynomials and some of their properties. In particular, using degenerate Appell polynomials, we derive the differential equations satisfied by certain degenerate complex polynomials based on the quasi-monomiality principle.
The paper is organized as follows. In
Section 2, we introduce degenerate complex Appell polynomials with cosine- and sine-Appell polynomials and prove some of their properties and relations.
Section 3 presents illustrative examples of differential equations satisfied by degenerate complex Appell polynomials. Finally,
Section 4 concludes the paper.
2. Degenerate Complex Appell Polynomials
In this section, we introduce degenerate complex Appell polynomials based on degenerate generating functions and study some of their properties. Furthermore, cosine- and sine-Appell polynomials are presented by splitting the complex Appell polynomials into their real ℜ and imaginary ℑ parts. We first provide the definition of degenerate Appell polynomials.
Definition 1. We define the sequence of degenerate complex Appell polynomials using the generating functionwhere denotes the degenerate Appell numbers. Here, the degenerate generating functions , which correspond to in the Appell polynomials (1), are obtained by the degenerate exponential function , which corresponds to the standard exponential function in (see, for example, Table 1). Remark 1. 1. Clearly, the first degenerate Appell number is nonzero and is independent of λ as by definition (ii);
2. The sequence satisfies conditions (i) and (ii);
3. For , the sequence satisfiesas the identity for can be obtained by the binomial convolution of the sequences and . On the basis of the degenerate complex Appell polynomials, we split complex values into real and imaginary parts using Euler’s formula.
Definition 2. For a nonnegative integer n, we define the degenerate cosine-Appell polynomials and the degenerate sine-Appell polynomials by the following generating functions: By Definitions 1 and 2, it follows that for
,
The degenerate trigonometric functions and have the following property.
Theorem 1. For and , and in (3) can be represented in terms of and as follows: Proof. The proof follows easily by comparing the coefficients in the polynomial expansion in (1) and (2), and by using Euler’s formula . □
Remark 2. It is noted that the sequences and can be explicitly determined when is given. For example, for and as defined in Table 1, the first five polynomials can be listed as in Table 2 and Table 3. One can verify that from [7], , , , and . Moreover, Equation (4) holds. We now provide an expression of and as well as some of their properties.
Theorem 2. Let n be a non-negative integer. Then, the following identities hold: where represents the greatest integer less than or equal to n.
Proof. By considering the product of and , the identity for is easily obtained by the binomial convolution of the sequences and . Similarly, the identity for can be obtained by the binomial convolution of the sequences and . □
Theorem 3. If we let for some sequence , then the followings hold: Proof. We only prove the first formula, as the proof of the second is similar. We consider the product
. First, using the binomial convolution of the sequences
and
, we have that
is the exponential generating function of the sequence
Then, by rewriting the product
as
and using the binomial convolution of the sequences
and
, we obtain that
is the exponential generating function of the sequence
Thus, the first identity is proved. □
Before providing an example of Theorem 3, we note that the
-falling factorial sequence
is given by the degenerate exponential function as follows [
10,
15]:
where
Example 1. Type 2 degenerate Euler polynomials are defined by the generating functions One can verify that the identities in Theorem 3 hold because .
We now show that the degenerate cosine- and sine-Appell polynomials ( and ) are represented by the Stirling numbers of the second kind .
Theorem 4. For , the degenerate cosine- and sine-Appell polynomials satisfy Proof. We first note that
is the exponential generating function of the sequence
. Then, considering the product
and
, the binomial convolution of the sequences
and
implies that
is the exponential generating function of the sequence
Similarly, by the binomial convolution of the sequences
and
, we have that
is the exponential generating function of the sequence
□
Recalling that
(see [
3,
16]), where
and
are the Stirling numbers of the second kind and the falling factorial, respectively, we have the following corollary.
Corollary 1. For , the following identities hold: Example 2. 1. If , then we have the sequence of the degenerate complex Euler polynomials in Table 1. Thus, the degenerate cosine- and sine-Euler polynomials can be obtained by 2. When , we obtain the sequence of the degenerate complex Bernoulli polynomials in Table 1. Hence, the degenerate cosine- and sine-Bernoulli polynomials are given by Remark 3. In particular, when in Definition 1, we consider the cosine- and sine- sequences and given by (see [8]) Then, as for in Theorem 2, we have that The following two theorems show that the degenerate complex Appell polynomials can be split into and .
Theorem 5. For , the degenerate complex Appell polynomials are related to and as follows: Proof. As
is the exponential generating functions for
, we immediately have (
5) by the binomial convolution of the sequences
and
, as
. □
Theorem 6. For , the degenerate cosine- and sine-Appell polynomials have the following properties: Proof. Let us consider the trigonometric identities
We note that
and
in Remark 3. As
is the exponential generating function for
, we obtain (
6) by the binomial convolutions. Equation (
7) can be proved similarly. □
Theorem 7. For , the following identities for degenerate cosine- and sine-Appell polynomials hold: Proof. As
is the exponential generating function for
, we obtain (
8) by the binomial convolution of the sequences
and
. Similarly, (
9) can be proved. □
As is obtained by the binomial convolution of the sequences and , we obtain the following corollary.
Corollary 2. For , the following identities for and hold: 3. Illustrative Examples of Differential Equations
In this section, we present two examples of degenerate complex Appell polynomials: the degenerate Bernoulli and Euler polynomials. By using quasi-monomiality, we derive the related differential equation satisfied by each of these polynomials. To this end, we first recall the definition of quasi-monomiality and some related properties.
By [
3,
17], a polynomial set
is called quasi-monomial if there exist two linear operators
and
such that
where
and
are called the derivative and multiplication operators, respectively.
On the basis of the results related to the monomiality principle, the quasi-monomial polynomials
satisfy the following differential equation:
In particular, as the sequence
is Appell-type, we have (more details are in [
3]):
or equivalently,
3.1. Degenerate complex Bernoulli polynomials
The degenerate complex Bernoulli polynomials
are defined by the degenerate generating function
. That is,
By simple computations, it is easily seen that the following expansion holds true:
where the first six coefficients are the following:
As the operators for the degenerate complex Bernoulli polynomials
are given by
the degenerate complex Bernoulli polynomials satisfy the following differential equation:
We note that for the degenerate complex Bernoulli polynomials
of degree
n, the differential equation is equivalently expressed by
3.2. Degenerate Complex Euler Polynomials
The degenerate complex Euler polynomials
are given by the degenerate generating function
. Equivalently,
The expansion for
is given by
where the first six coefficients are the following:
As the operators for the degenerate complex Euler polynomials
are given by
the differential equation of the degenerate complex Euler polynomials is
It is noted that for the degenerate complex Euler polynomials
of degree
n, the differential equation is equivalently given by