#### Abstract

The aim of this study is to represent any polynomial in terms of the degenerate Genocchi polynomials and more generally of the higher-order degenerate Genocchi polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

#### 1. Introduction and Preliminaries

In this study, we derive formulas expressing any polynomial in terms of the degenerate Genocchi polynomials ((14)) with the help of umbral calculus (Theorem 1) and illustrate our results with some examples (Section 5). This can be generalized to the higher-order degenerate Genocchi polynomials ((15)). Indeed, we deduce formulas for representing any polynomial in terms of the higher-order degenerate Genocchi polynomials again by using umbral calculus (Theorem 2). The contribution of this study is the derivation of such formulas which have potential applications to finding some interesting polynomial identities, as illustrated in Section 5.

Many interesting identities have been derived by using similar formulas for representations by Bernoulli, Euler, and Frobenius-Euler polynomials (see ). The list in the references is far from being exhaustive. However, the interested reader can easily find more related studies in the literature. Also, we should mention here that there are other ways of obtaining the same result as the one in (2). One of them is to use Fourier series expansion of the function obtained by extending by periodicity of period 1 of the polynomial function restricted to the interval .

Let . Write , where are the Bernoulli polynomials defined by . Then, it is known that

Applying the formulas in (1) to the polynomial , we can get an identity which yields, after slight modification, the following identity:where , and .

Letting in (2), we obtain a slight variant of the well-known Miki’s identity:

Letting in (2), we get the Faber–Pandharipande–Zagier (FPZ) identity:where , for all .

Here, it should be stressed that the other proofs of Miki’s () and FPZ identities ([14,15]) are quite involved, while our proofs of Miki’s and Faber–Pandharipande–Zagier identities follow from the simple formulas in (1) involving only derivatives and integrals of the given polynomials.

The outline of this study is as follows. In Section 1, we recall some necessary facts that are needed throughout this study. In Section 2, we go over umbral calculus briefly. In Section 3, we derive formulas expressing any polynomial in terms of the degenerate Genocchi polynomials. In Section 4, we derive formulas representing any polynomial in terms of the higher-order degenerate Genocchi polynomials. In Section 5, we illustrate our results with some examples. Finally, we conclude our study in Section 6.

The Euler polynomials are defined by

When , are called the Euler numbers. We observe that . The first few terms of are given by

More generally, for any nonnegative integer , the Euler polynomials of order are given by

The Genocchi polynomials are defined by

When , are called the Genocchi numbers. We observe that ,,, and , for . Note that . The first few terms of are given by

More generally, for any nonnegative integer , the Genocchi polynomials of order are given by

Observe here that and that has degree , for .

For any nonzero real number , the degenerate exponentials are given by ([3, 5, 6])

Carlitz  introduced a degenerate version of the Euler polynomials , called the degenerate Euler polynomials and denoted by , which are given by

For , are called the degenerate Euler numbers.

More generally, for any nonnegative integer , the degenerate Euler polynomials of order are given by

A degenerate version of the Genocchi polynomials , called the degenerate Genocchi polynomials and denoted by , are given by ()

For , are called the degenerate Genocchi numbers. Note that .

More generally, for any nonnegative integer , the degenerate Genocchi polynomials of order are given by ()

For , are called the degenerate Genocchi numbers of order . Observe that and that has degree , for .

We remark that and , as tends to 0.

We recall some notations and facts about forward differences. Let be any complex-valued function of the real variable . Then, for any real number , the forward difference is given by

If , then we let

We also need

It is necessary to note that

In general, the th order forward differences are given by

For , we have

It is easy to see that

Finally, we recall that the Stirling numbers of the second kind are given by

#### 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 . Let be the field of complex numbers. Then, denotes the algebra of formal power series in over given byand indicates the algebra of polynomials in with coefficients in .

The set of all linear functionals on is a vector space as usual and denoted by . Let denote the action of the linear functional on the polynomial .

For with , we define the linear functional on by

From (25), we note thatwhere is Kronecker’s symbol.

Some remarkable linear functionals are as follows:

Let

Then, by (25) and (28), we get

That is, , as linear functionals on , shows the map from onto is one-to-one. Additionally, the map is linear and also onto in view of (25) and (28). Thus, it is a vector space isomorphism from onto .

Henceforth, denotes both the algebra of formal power series in 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 , the differential operator on is defined by

Extending (30) linearly, any power seriesgives the differential operator on defined by

It should be observed that, for any formal power series and any polynomial , we have

Here, we note that an element of is a formal power series, a linear functional, and a differential operator. Some notable differential operators are as follows:

The order of the power series is the smallest integer for which does not vanish. If , then is called an invertible series. If , then is called a delta series.

For with and , there exists a unique sequence (deg ) of polynomials, such that

The sequence is said to be the Sheffer sequence for , which is denoted by . We observe from (35) thatwhere .

In particular, if , then , and hence,

It is well known that if and only iffor all , where is the compositional inverse of , such that .

The following equations (39)–(41) are equivalent to the fact that is Sheffer for , for some invertible :with ,

For and , we havewhere

#### 3. Representation by Degenerate Genocchi Polynomials

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

From (11), we first observe that

From (39), we note that

It is immediate to see from (14) that

Now, we assume that has degree , and write . Let . Then, from (45) and (46), we have

For , from (45) and (47), we obtain

Letting in (48), we finally get

An alternative expression of (49) is given bysince .

From (50), we have another alternative expression of (49) which is given by

From (49), we get yet another expression of as follows:where .

Finally, from (49)–(52), we get the following theorem.where .

Theorem 1. Let . Then, we have , where ’s are given by the following various expressions:

Remark 1. Let . Write . As tends to 0, . Thus, we get the following result, namely, we have

Remark 2. The formulas in (1) and (54) and analogous formula for representation by Euler polynomials have been applied to many polynomials in order to obtain interesting identities for certain special polynomials and numbers. Some of the polynomials that have been considered are as follows:(a)Where the sum is over all nonnegative integers , such that , and are the nonnegative integers with .(b)Where the sum is over all nonnegative integers , such that , and are the nonnegative integers with .(c)Where the sum is over all positive integers and nonnegative integers , such that , and are the nonnegative integers with .

#### 4. Representation by Higher-Order Degenerate Genocchi Polynomials

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

As we noted in (45), we have , with . Let .

It is immediate to see from (15) that

Repeated application of (58) gives us

Now, we assume that has degree , and write . Then, from (59), we observe that

By using (45) and (60), for , we observe that

By evaluating (61) at , we obtain

Several alternative expressions of (62) follow from (20) and (22), which are given by

Another expression for follows from (23):

Summarizing the results so far, from (62) and (64), we obtain the following theorem.where ’s are given by the following various expressions:

Theorem 2. Let . Let . Then, we have

Remark 3. Let . Write . As tends to 0, . Thus, from Theorem 2, we have the following result.

#### 5. Examples

(a)Here, we illustrate Theorem 1, with . Let .For with , we haveThus, from (68), we have(b)Here, we illustrate Theorem 1, for . For this, we first recall from  thatWhere is the harmonic number. Let . For , with , we obtainBy proceeding as we did in (a), from (71), we see thatThus, from (72) and for , we have(c)In , it is shown that the following identity holds for :Where is the harmonic number.Write .By proceeding similarly to (b), we obtain the following identity:(d)In , it is proved that the following identity is valid for :Again, by proceeding analogously to (b), we get the following identity:(e)Nielsen [21, 22] expressed the product of a Bernoulli polynomial and an Euler polynomial in terms of Euler polynomials, namely, for nonnegative integers and ,Again, in a similar way to (b), we can show thatWe also represent in terms of the degenerate Genocchi polynomials of order . Indeed, from Theorem 2, we obtain three expressions:Where the innermost sum in (80) is over all integers and satisfying , and .(f)In (4.20) of , it is shown that the following identity holds for :Then, proceeding similarly to (b), we can show thatIn general, in terms of the degenerate Genocchi polynomials of order , we obtain the following: