1. Introduction
Appell polynomials are very frequently used in various problems in pure and applied mathematics related to functional equations in differential equations, approximation theories, interpolation problems, summation methods, quadrature rules, and their multidimensional extensions (see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]). Also, for a further general account of the study in Appell polynomials, a number of applications can be found (see [
4] and references therein).
The sequence of complex Appell polynomials
can be obtained by either of the following equivalent conditions:
or the following formal equality
where
is a formal power series with coefficients
called by Appell numbers.
There is a large number of classical sequences of polynomials in Appell polynomials and a list of famous classical Appell polynomials is shown as the Bernoulli polynomials, the Euler polynomials, the Hermite polynomials, the Genocchi polynomials, the generalized Bernoulli polynomials, the generalized Euler polynomials, etc. (see [
2,
3] for more examples).
Many authors have obtained useful results by considering the Appell polynomials of a complex variable by splitting complex-valued polynomials into real and imaginary values: Analytic properties of the sequence of complex Hermite polynomials are studied in [
12] and their orthogonality relation is established in [
13,
14]. Also, authors in [
15] show the representation of the real and imaginary parts in the complex Appell polynomials in terms of the Chebyshev polynomials of the first and second kind.
With the help of the research of complex Appell polynomials, their degenerate versions have been also extensively studied looking for useful identities, as well as their related properties, since Carlitz introduced degenerate formulas of special numbers and polynomials in [
16,
17]. Further, there have been studies of various degenerate numbers and polynomials by means of degenerate types of generating functions, combinatorial methods, umbral calculus, and differential equations. For example, several authors have studied the degenerate types of Appell polynomials, such as Bernoulli and Euler polynomials (see [
18,
19,
20,
21,
22,
23]) and their complex version [
24], degenerate gamma functions, degenerate Laplace transforms [
25], and their modified ones [
26].
The research for degenerate versions of known special numbers and polynomials brought many valuable identities and properties into mathematics. In the future, we hope the results of the degenerate types of complex Appell polynomials can be further applicable to many different problems in various areas.
The aim of this paper is to introduce Appell polynomials of a complex variable and their degenerate formulas and provide some of their properties and examples. Also, we study some further properties of the degenerate type of Appell polynomials and show that degenerate cosine- and sine-Appell polynomials can be expressed by the Stirling numbers of the first kind.
The paper is organized as follows. In
Section 2, we recall the complex Appell polynomials with cosine- and sine-Appell polynomials and present some properties and their relations.
Section 3 introduces the degenerate version of complex Appell polynomials and provides some expressions, properties, and examples. Finally,
Section 4 contains the conclusion of this study.
2. Complex Appell Polynomials
In this section, we introduce the cosine-Appell polynomials and sine-Appell polynomials by splitting complex Appell polynomials into real ℜ and imaginary ℑ parts, and present some properties, which can apply to any Appell-type polynomial, as mentioned in the introduction.
Definition 1. For , we define the cosine-Appell polynomials and the sine-Appell polynomials by the generating functions respectively, The definition in (
3) with the fact (
2) implies that
Also, it is easily observed that for
,
which show
and
Further, noting that , for , it can be checked that the cosine-Appell polynomials and the sine-Appell polynomials satisfy the following properties:
- (i)
- (ii)
- (iii)
and
- (iv)
- (v)
- (vi)
The above properties are easily proved by the comparison of coefficients after polynomial expansion of the generating functions and we omit the proofs here for lack of space.
We next investigate further properties of complex Appell polynomials.
Theorem 1. For , , the following product of complex Appell polynomials is established: Proof. The product of the identities for
and
from (
4) shows the desired identity. □
Corollary 1. For , , we have the identity, Lemma 1. For , , the real ℜ and imaginary ℑ parts of complex Appell polynomials satisfy that Proof. Considering that
, we have
Thus, using identity for the binomials convolution of sequences
and
, we get identity (
5). Similarly, one can show identity (6) by considering
. □
Remark 1. Note that the sequences and can be explicitly determined when is given, namely, complex Euler polynomials for and complex Bernoulli polynomials for ,andrespectively. For example, the first four consecutive polynomials are listed as in Table 1 and Table 2. Lemma 2. Let n be a nonnegative integer and . Then, the complex Appell polynomials satisfy the following identities: Proof. The right side of the first identity we get directly by the 2-fold binomial convolution of sequence (thus using the square of exponential generating function ). Alternatively, rewriting as the product of two exponential generating functions by this way: , we obtain the left side of the first identity by the binomial convolution of sequences and . The second identity we obtain in a similar argument. □
Remark 2. In particular, if we consider in Definition 1, Equation (3) shows thatandfor some sequences and . As for , we have from identities (5) and (6) The sequences and satisfy the following formulas.
Theorem 2. For , the following identities are established: Proof. If we denote
,
, the trigonometric identities yield that
As the right hand side of (
11) exponential generating functions for
, we have the first formula (
9) by the difference between the binomial convolutions of sequences
and
and the binomial convolutions of sequences
and
. The second formula (10) we get in a similar way by using (
12). □
The following two subsequent theorems show that the complex Appell polynomials can be split into and and their relations.
Theorem 3. For , the Appell-type polynomials satisfy the following relations with and , Proof. As
exponential generating functions for
,
, we have directly equation (
13) by the binomial convolution of sequences
and
. □
Theorem 4. For , the cosine-Appell polynomials and the sine-Appell polynomials satisfy the following properties, Proof. As
exponential generating function for
, we get the first line in formula (
14) by the binomial convolution of sequences
and
. The second line of (
14) follows from formula (
5). Similarly, identity (
15) can be proved. □
Next, the derivatives of
and
show that the sequence
satisfies the condition (
1).
Theorem 5. For all and , the sequence is verified by a sequence of complex Appell polynomials in terms of and .
Proof. Noting that
, the derivative of the cosine-Appell polynomials satisfies
which implies that
Similarly, it can be seen that
By using (
16) and (
17), it is easily shown that
□
3. Degenerate Type of Complex Appell Polynomials
In this section, we introduce the degenerate type of complex Appell polynomials based on the non-degenerate ones given in Definition 1 and study some of their properties. To do this, we first recall and introduce several definitions, some notations, and basic properties.
Let us recall several definitions: the degenerate exponential function
for
is defined by (see [
27,
28,
29])
It is noted that
.
The falling factorial sequence is defined by (see [
30,
31,
32,
33,
34])
Similarly, the
-falling factorial sequence is given by (see [
27])
Then, it can be easily checked that the factorial sequences in (
19) and (
20) satisfy
and
for all
.
Further, the
- binomial expansion is defined by (see [
27])
in which the
-binomial coefficient satisfies
Definition 2. Let us assume that for some sequence and satisfy . Then we define the degenerate type of complex Appell polynomials by the generating function Remark 3. If is considered as and for , then one can have the sequences and of degenerate types of complex Euler polynomials and complex Bernoulli polynomialsandrespectively. Letting
in Equation (
18), we find, for
and we can formulate the following degenerate Euler formula given by
where
Thus, the complex value of the degenerate exponential function is split into the real and imaginary values.
Similarly,
, we can put
where
so that
Now, using the degenerate functions (
26), we define the following polynomials.
Definition 3. For a nonnegative integer n, let us define the degenerate cosine-Appell polynomials and the degenerate sine-Appell polynomials by the generating functions, respectively, as follows: From Definitions 2 and 3 with property (
21), one can see that for
,
Also, the following property can be stated.
Lemma 3. For and , let and be the degenerate cosine-Appell and sine-Appell polynomials defined in (27). Then, we have Proof. We first note that from (
26) and (
27),
and
, since
and
. Thus,
from (
28), so that the desired identities are easily obtained. □
Remark 4. It is noted that the sequences and can be explicitly determined when is specified. For example, for and as defined in (23) and (24) respectively, the first four consecutive polynomials can be listed as in Table 3 and Table 4. One can check that from Table 1 and Table 2, , , , and . Lemma 4. Let n be a non-negative integer. The degenerate cosine and sine functions, namely and , are the exponential generating functions of the sequencesandrespectively, where are the Stirling polynomials of the first kind, which satisfy (for details see [35,36,37]) Proof. We show the proof for
only, as the proof for
can be done similarly:
where we use the well-known identity (see [
35,
37])
□
We next give an expression of and in terms of Stirling numbers of the first kind.
Theorem 6. Let n be a non-negative integer. Then, the following identities hold: Proof. The identity for
we get easily by the binomial convolution of the sequence
and the sequence in Formula (
29). Similarly, the identity for
we obtain by the binomials convolution of sequences
and (
30). □
Lemma 5. If we assume that for some sequence , then for , we have Proof. We show the proof of the first formula only, as the proof of the second one can be done similarly. We will consider product
. Firstly, using the binomial convolution of sequences
and (
29), we have that
is the exponential generating function of the sequence
Secondly, by rewriting the product
as
and using the binomial convolution of sequences
and
, we get that
is the exponential generating function of the sequence
thus, the first identity is proved. □
Example 1. Type 2 degenerate Euler polynomials are defined by the generating functions (see [26]) One can check that the identities in Lemma 5 are established, considering that .
Finally, we show that the degenerate types of cosine- and sine-Appell polynomials, and , are represented by the Stirling numbers of the first kind .
Theorem 7. For , the degenerate type of cosine-Appell polynomials is satisfied by In particular, for , the degenerate type of sine-Appell polynomials holds true: Proof. We prove the first identity only, as the second one can be proved similarly. By the binomials convolution of sequences
and (
29) and using identity (
28) we obtain the first identity. □
Example 2. - 1.
If , then we have the sequence of the degenerate Euler polynomials defined in (23). Thus, the degenerate cosine-Euler polynomials and sine-Euler polynomials can be obtained by (see [22] (Theorem 3)replacing by in Theorem 7. - 2.
When , the sequence of degenerate Bernoulli polynomials defined in (24) can be considered. Hence, the degenerate cosine-Euler polynomials and sine-Euler polynomials can be obtained by (see [22] (Theorem 7))replacing by in Theorem 7.
4. Conclusions
In this paper, we study the general properties and identities of the complex Appell polynomials by treating the real and imaginary parts separately, which provide the cosine-Appell and sine-Appell polynomials. These presented results can be applied in any complex Appell-type polynomials such as complex Bernoulli polynomials and complex Euler polynomials. Further, we consider the degenerate version of complex Appell polynomials by splitting them into the degenerate cosine-Appell and sine-Appell polynomials and present some of their properties, relations, and examples. Finally, we show that any degenerate cosine-Appell- and sine-Appell-type polynomial can be expressed in terms of the Stirling numbers of the first kind.