Representations by degenerate Daehee polynomials

: In this paper, we consider the problem of representing any polynomial in terms of the degenerate Daehee polynomials and more generally of the higher - order degenerate Daehee polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.


Introduction and preliminaries
The aim of this paper is to derive formulas (see Theorem 3.1) expressing any polynomial in terms of the degenerate Daehee polynomials (see (1.12)) with the help of umbral calculus and to illustrate our results with some examples (see Chapter 6). This can be generalized to the higher-order degenerate Bernoulli polynomials (see (1.13)). Indeed, we deduce formulas (see Theorems 4.1) for representing any polynomial in terms of the higher-order degenerate Daehee polynomials again by using umbral calculus. Letting → λ 0, we obtain formulas (see Remarks 3.2 and 4.2) for expressing any polynomial in terms of the Daehee polynomials (see (1.10)) and of the higher-order Daehee polynomials (see (1.11)). These formulas are also illustrated in Chapter 5. The contribution of this paper is the derivation of such formulas that, we think, have many potential applications. Let are the Bernoulli polynomials (see (1.3)). Then, it is known (see [1]) that  The following identity (see [1,2]) is obtained by applying the formula in Letting = x 0 and = x 1 2 in (1.2), respectively, give a slight variant of Miki's identity and the Faber-Pandharipande-Zagier (FPZ) identity. Here, it should be emphasized that the other proofs of Miki's (see [3][4][5]) and FPZ identities (see [6,7]) are quite involved, while our proofs of Miki's and FPZ identities follow from the simple formula in (1.1) involving only derivatives and integrals of the given polynomials.
Analogous formulas to (1.1) can be obtained for the representations by Euler, Frobenius-Euler, ordered Bell and Genocchi polynomials. Many interesting identities have been derived by using these formulas (see [1,[8][9][10][11][12][13][14] and references therein). The list in the references is far from being exhaustive. However, the interested reader can easily find more related papers in the literature. Also, we should mention here that there are other ways of obtaining the same result as the one in (1.2). One of them is to use Fourier series expansion of the function obtained by extending by periodicity 1 of the polynomial function restricted to the interval [ ) 0, 1 (see [2,15,16]). The outline of this paper is as follows. In Section 1, we recall some necessary facts that are needed throughout this paper. In Section 2, we go over umbral calculus briefly. In Section 3, we derive formulas expressing any polynomial in terms of the degenerate Daehee polynomials. In Section 4, we derive formulas representing any polynomial in terms of the higher-order degenerate Daehee polynomials. In Section 5, we illustrate our results with examples of representation by the Daehee polynomials. In Section 6, we illustrate our results with examples of representation by the degenerate Daehee polynomials. Finally, we conclude our paper in Section 7.
are called the Euler numbers. We observe that ( ) The first few terms of E n are given by: The first few terms of G n are given by: For any nonzero real number λ, the degenerate exponentials are given by Here, we recall that the λ-falling factorials are given by

8)
Especially, ( ) ( ) = x x n n ,1 are called the falling factorials and hence given by The compositional inverse of ( ) e t λ is called the degenerate logarithm and given by  are called the degenerate Daehee numbers and introduced in [7] (see also [14]).
More generally, for any nonnegative integer r, the degenerate Daehee polynomials which are degenerate versions of the Daehe polynomials of order r in (1.11). We remark that , as λ tends to 0.
We recall some notations and facts about forward differences. Let f be any complex-valued function of the real variable x. Then, for any real number a, the forward difference Δ a is given by In general, the nth oder forward differences are given by Finally, we recall that the Stirling numbers of the second kind ( ) S n k , 2 can be given by means of

Review of umbral calculus
Here, we will briefly go over very basic facts about umbral calculus. For more details on this, we recommend the reader to refer to [3,20,22]. Let be the field of complex numbers. Then, denotes the algebra of formal power series in t over , given by where c is a complex number.
where δ n k , is the Kronecker's symbol. Some remarkable linear functionals are as follows: Then, by (2.1) and (2.3), we obtain denotes both the algebra of formal power series in t and the vector space of all linear functionals on .
is called the umbral algebra and the umbral calculus is the study of umbral algebra. For each nonnegative integer k, the differential operator t k on is defined by Extending (2.4) linearly, any power series gives the differential operator on defined by It should be observed that, for any formal power series ( ) f t and any polynomial ( ) p x , we have Here, we note that an element ( ) f t of is a formal power series, a linear functional, and a differential operator. Some notable differential operators are as follows: The order ( ( )) o f t of the power series ( )( ) ≠ f t 0 is the smallest integer for which a k does not vanish.
n , then ( ) = p x x n n , and hence, , for some invertible ( ) g t : be sequences of polynomials. Then, the umbral composition of ( ) q x n with ( ) p x n is defined to be the sequence

Representations by degenerate Daehee polynomials
Our interest here is to derive formulas expressing any polynomial in terms of the degenerate Daehee polynomials.
n λ n , x has degree n, and write ( ) . Then, from (3.5), we have

7)
Letting = x 0 in (3.7), we finally obtain Now, we want to find more explicit expressions for (3. where ( ) We note from (3.5) and (3.9), in passing, that the following holds: By making use of (1.17) and (3.10), an alternative expression of (3.10) is given by We obtain yet another expression from (1.18), (3.8), and (3.9), which is given by where we need to note that has degree n.
We observe that

Examples on representation by Daehee polynomials
Here, we illustrate our formulas in Remarks 3.  Thus, we have the following: where we understand that the sum in (5.5) is zero for = − k n 1 or n. Thus, we obtain the following identity: (c) In [12], it is shown that the following identity holds for ≥ n 2: (d) In [16], it is proved that the following identity is valid for ≥ n 2: Therefore, we obtain the following identity: (e) Nielsen [2,19] also represented products of two Euler polynomials in terms of Bernoulli polynomials as follows: