## Abstract

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.

## 1 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

Let

The following identity (see [1,2]) is obtained by applying the formula in (1.1) to the polynomial

where

Letting

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

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.

The Bernoulli polynomials

When

More generally, for any nonnegative integer

When

The Euler polynomials

When

The Genocchi polynomials

When

For any nonzero real number

Here, we recall that the

Especially,

The compositional inverse of

which satisfies

Note here that

Recall that the Daehee polynomials

When

More generally, for any nonnegative integer

When

The degenerate Daehee polynomials

which are degenerate versions of the Daehee polynomials in (1.10). For

More generally, for any nonnegative integer

which are degenerate versions of the Daehe polynomials of order

We recall some notations and facts about forward differences. Let

If

In general, the

For

Finally, we recall that the Stirling numbers of the second kind

## 2 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

and

Let

where

For

From (2.1), we note that

where

Some remarkable linear functionals are as follows:

Let

Then, by (2.1) and (2.3), we obtain

That is,

Henceforth,

Extending (2.4) linearly, any power series

gives the differential operator on

It should be observed that, for any formal power series

Here, we note that an element

The order

For

The sequence

where

In particular, if

It is well known that

for all

Equations (2.12)–(2.14) are equivalent to the fact that

with

Let

## 3 Representations by degenerate Daehee polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the degenerate Daehee polynomials.

From (1.7), (1.9), and (1.11), we first observe that

From (1.15), (2.7), (2.8), (2.12), (3.1), and (3.2), we note that

Now, we assume that

For

Letting

Now, we want to find more explicit expressions for (3.8). As

From (2.7), (2.15), and (3.1), noting that

where

We note from (3.5) and (3.9), in passing, that the following holds:

From (2.7) and (3.9), we deduce

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

Finally, from (3.10)–(3.12), and (3.8), we obtain the following theorem.

## Theorem 3.1

*Let*
*with*
*Then, we have*
*where*

*where*
*and*
*denotes the umbral composition of*
*with*

## Remark 3.2

Let

## 4 Representations by higher-order degenerate Daehee polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the higher-order degenerate Daehee polynomials.

With

From (1.15), (2.7), (2.8), (2.12), (4.1), and (4.2), we note that

Now, we assume that

For

Letting

This also follows from the observation

Now, we want to find more explicit expressions for (4.8). As

From (2.7), (2.15), and (4.1), noting that

where

We note from (4.5) and (4.9), in passing, that the following holds:

From (2.7) and (4.9), we deduce

By making use of (1.17) and (4.10), an alternative expression of (3.10) is given by

We obtain yet another expression from (1.18), (4.8), and (4.9), which is given by

where we need to observe that

Finally, from (4.10)–(4.12) and (4.8), we obtain the following theorem.

## Theorem 4.1

*Let*
*with*
*Then, we have*
*where*

*where*
*indicates the umbral composition of*
*with*
*and*
*denotes the linear integral operator given by*

We observe that

## Remark 4.2

Let

We note that

## 5 Examples on representation by Daehee polynomials

Here, we illustrate our formulas in Remarks 3.2 and 4.2 with some examples.

(a) Let

which are well known.

Thus, we obtain the following identity:

Next, we let

Now, by making use of Remark 4.2, we obtain

Thus, we have the following:

(b) Here, we consider

where

where we understand that the sum in (5.5) is zero for

(c) In [12], it is shown that the following identity holds for

where

Write

By proceeding similarly to (b), we see that

Thus, (5.7) implies the next identity:

(d) In [16], it is proved that the following identity is valid for

Again, by proceeding analogously to (b), we can show that

Therefore, we obtain the following identity:

(e) Nielsen [2,19] also represented products of two Euler polynomials in terms of Bernoulli polynomials as follows:

In the same way as (b), we can show that

Thus, we arrive at the next identity:

## 6 Examples on representation by degenerate Daehee polynomials

Here, we illustrate our formulas in Theorems 3.1 and 4.1.

(a) Let