A Note on the Degenerate Type of Complex Appell Polynomials

: In this paper, complex Appell polynomials and their degenerate-type polynomials are considered as an extension of real-valued polynomials. By treating the real value part and imaginary part separately, we obtained useful identities and general properties by convolution of sequences. To justify the obtained results, we show several examples based on famous Appell sequences such as Euler polynomials and Bernoulli polynomials. Further, we show that the degenerate types of the complex Appell polynomials are represented in terms of the Stirling numbers of the ﬁrst kind.

The sequence of complex Appell polynomials {A n (z)} ∞ n=0 can be obtained by either of the following equivalent conditions: or the following formal equality where n! , (a 0 = 0) is a formal power series with coefficients a n 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 Symmetry 2019, 11, 1339; doi:10.3390/sym11111339www.mdpi.com/journal/symmetryestablished 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.

Complex Appell Polynomials
In this section, we introduce the cosine-Appell polynomials and sine-Appell polynomials by splitting complex Appell polynomials into real and imaginary parts, and present some properties, which can apply to any Appell-type polynomial, as mentioned in the introduction.Definition 1.For n ∈ N ∪ {0}, we define the cosine-Appell polynomials A (c) n (x, y) and the sine-Appell polynomials A (s) n (x, y) by the generating functions respectively, The definition in (3) with the fact (2) implies that Also, it is easily observed that for z = x − iy, and Further, noting that n (x, y) = (A n (z)), z = x + iy for n ≥ 0, it can be checked that the cosine-Appell polynomials and the sine-Appell polynomials satisfy the following properties: 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 n, m ∈ N ∪ {0}, z = x + iy, the following product of complex Appell polynomials is established: m (x, −y) Proof.The product of the identities for A n (z) and A m ( z) from (4) shows the desired identity.
Remark 1.Note that the sequences {A and respectively.For example, the first four consecutive polynomials are listed as in Tables 1 and 2.
Table 1.Expressions of the first four Let n be a nonnegative integer and z = x + iy.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 {A n (x)} ∞ n=0 (thus using the square of exponential generating function (A(t)e xt ) 2 ).Alternatively, rewriting (A(t)e xt ) 2 as the product of two exponential generating functions by this way: (A(t)e (x+iy)t )(A(t)e (x−iy)t ), we obtain the left side of the first identity by the binomial convolution of sequences {A n (z)} and {A n ( z)}.The second identity we obtain in a similar argument.

Remark 2. In particular, if we consider
for some sequences C n (z) and S n (z).As A n (x) = x n for A(t) = 1, we have from identities (5) and (6) The sequences C n (z) and S n (z) satisfy the following formulas.
The following two subsequent theorems show that the complex Appell polynomials can be split into C n (z) and S n (z) and their relations.Theorem 3.For n ∈ N ∪ {0}, the Appell-type polynomials satisfy the following relations with C n (z) and S n (z), Proof.As A(t)e xt (cos(yt) + i sin(yt)) exponential generating functions for A n (z), z = x + iy, we have directly equation ( 13) by the binomial convolution of sequences {a n } ∞ n=0 and {C n (z) + iS n (z)} ∞ n=0 .
Theorem 4. For k > 0, the cosine-Appell polynomials and the sine-Appell polynomials satisfy the following properties, Proof.As A(t)e kt e ±xt cos(±yt) exponential generating function for n (k ± x, y), we get the first line in formula ( 14) by the binomial convolution of sequences {A n (k)} ∞ n=0 and {C n (x + iy)(±1) n } ∞ n=0 .The second line of (14) follows from formula (5).Similarly, identity (15) can be proved.

Next, the derivatives of A
Similarly, it can be seen that By using ( 16) and ( 17), it is easily shown that

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.
Then we define the degenerate type of complex Appell polynomials by the generating function respectively.
Letting x = i in Equation ( 18), we find, for λ ∈ R\{0} so that lim From Definitions 2 and 3 with property (21), one can see that for z = x + iy, Also, the following property can be stated.

Lemma 3.
For n ≥ 0 and z = x + iy, let A n (x, y; λ) and A n (x, y; λ) be the degenerate cosine-Appell and sine-Appell polynomials defined in (27).Then, we have Proof.We first note that from ( 26) and (27), 28), so that the desired identities are easily obtained.
n (x, y; λ)} ∞ n=0 can be explicitly determined when A λ (t) is specified.For example, for A n (z; λ) = E n (z; λ) and A n (z; λ) = B n (z; λ) as defined in (23) and (24) respectively, the first four consecutive polynomials can be listed as in Tables 3 and 4. One can check that from Tables 1 and 2 Let n be a non-negative integer.The degenerate cosine and sine functions, namely cos and    respectively, where S 1 (n, m) are the Stirling polynomials of the first kind, which satisfy (for details see [35][36][37]) Proof.We show the proof for cos where we use the well-known identity (see [35,37]) We next give an expression of A Theorem 6.Let n be a non-negative integer.Then, the following identities hold: Proof.The identity for A (c) n (x, y; λ) we get easily by the binomial convolution of the sequence {A n (x; λ)} ∞ n=0 and the sequence in Formula (29).Similarly, the identity for A n (x, y; λ) we obtain by the binomials convolution of sequences {A n (x; λ)} ∞ n=0 and (30).

Lemma 5. If we assume that
Proof.We show the proof of the first formula only, as the proof of the second one can be done similarly.
Finally, we show that the degenerate types of cosine-and sine-Appell polynomials, A n (x, y; λ) and A Proof.We prove the first identity only, as the second one can be proved similarly.By the binomials convolution of sequences {A n (x; λ)} ∞ n=0 and (29) and using identity (28) we obtain the first identity.

nTheorem 5 .
(x, y) and A (s) n (x, y) show that the sequence {A n (z)} ∞ n=0 satisfies the condition(1).For all n ∈ N ∪ {0} and z = x + iy, the sequence {A n (z)} ∞ n=0 is verified by a sequence of complex Appell polynomials in terms of A (c) n (x, y) and A (s) n (x, y).Proof.Noting that ∂ ∂x A (c) 0 (x, y) = 0, the derivative of the cosine-Appell polynomials satisfies e xt cos(yt) = A(t)te xt cos(yt) =

Definition 3 .
) = cos(yt) + i sin(yt).Now, using the degenerate functions(26), we define the following polynomials.For a nonnegative integer n, let us define the degenerate cosine-Appell polynomials A (c) n (x, y; λ) and the degenerate sine-Appell polynomials A (s) n (x, y; λ) by the generating functions, respectively, as follows:A λ (t)e x λ (t) cos

Remark 4 .
It is noted that the sequences {A (c) n (x, y; λ)} ∞ n=0 and {A

λ
(t), are the exponential generating functions of the sequences

λ
(t) only, as the proof for sin

n
(x, y; λ) and A (s) n (x, y; λ) in terms of Stirling numbers of the first kind.

λ
(t)  is the exponential generating function of the sequence

Example 1 .
first identity is proved.Type 2 degenerate Euler polynomials E

Table 2 .
Expressions of the first four B n (x, y) and B (s) n (x, y).

Table 3 .
Expressions of the first four E

Table 4 .
Expressions of the first four B