Classical Problems in Algebra and Number TheoryView this Special Issue
Representation by Degenerate Genocchi Polynomials
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 [1–9]). 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 ([11–13]) 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 .
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 [18–20]. 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:
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 .
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
Letting in (48), we finally get
An alternative expression of (49) is given bysince .
From (49), we get yet another expression of as follows: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 evaluating (61) at , we obtain
Another expression for follows from (23):
Theorem 2. Let . Let . Then, we have
Remark 3. Let . Write . As tends to 0, . Thus, from Theorem 2, we have the following result.