Various families of polynomials play a key role in applied mathematics due to the fact that they can be described in many different ways, for example, by orthogonality conditions, by generating functions, as solutions to differential equations, by integral transforms, by recurrence relations, by operational formulas, and so on. In light of their many important properties, their extensions and generalizations with applications are also considered by the researchers in mathematical and physical sciences. The resulting formulas are very important and potentially useful, because they include expansions for many transcendent expressions of mathematical physics in series of the classical orthogonal polynomials. The developments bear heavily upon the work of many researchers who have earlier studied the special polynomials with applications to *p*-adic analysis, *q*-analysis, umbral analysis, and so on (see, for example, the recent work [3–22] and [23]).

Frobenius [10] (see also [4]) studied the polynomials \(F_{n}(x|u)\) in great detail by means of the following exponential generating function:

$$ \sum_{n=0}^{\infty }F_{n}(x|u) \frac{t^{n}}{n!}=\frac{1-u}{e^{t}-u} e ^{xt}\quad \bigl(u \in \mathbb{C} \setminus \{1\}\bigr). $$

(1.1)

Several identities and characterizations of the Frobenius polynomials \(F_{n}(x|u)\) can be found in the works by Kim *et al.* [14–17]. In the case when \(u=-1\) in (1.1), it reduces to the following relationship with the Euler polynomials \(E_{n}(x)\):

$$ F_{n}(x|-1)=E_{n} (x ), $$

which are given in (1.4) below. Owing to their important properties, and in the honor of Frobenius, the polynomials \(F_{n}(x|u)\) are called the Frobenius–Euler polynomials.

These polynomials are expressed recursively, in terms of the Frobenius–Euler numbers defined by

$$ F_{n}(u):=F_{n}(0|u), $$

as follows:

$$ F_{n}(x|u)=\sum_{k=0}^{n} \binom{n}{k}F_{k}(u)x^{n-k}\quad ( n\geqq 0 ), $$

(1.2)

where the Frobenius–Euler numbers \(F_{n}(u)\) satisfy the following recurrence relation:

$$ F_{0}(u)=1\quad \text{and}\quad \bigl(F(u)+1\bigr)^{n}-F_{n}(u)= (1-u ) \delta _{n,0}, $$

(1.3)

by simply replacing \(F^{n}(u)\) by \(F_{n}(u)\), \(\delta _{n,k}\) being the Kronecker delta.

The classical Bernoulli polynomials \(B_{n}(x)\) and the classical Euler polynomials \(E_{n}(x)\) are analogous to the Frobenius–Euler polynomials \(F_{n}(x|u)\). They are specified by the following exponential generating functions:

$$ \sum_{n=0}^{\infty }B_{n}(x) \frac{t^{n}}{n!}=\frac{t}{e^{t}-1} e ^{xt}\quad \text{and}\quad \sum_{n=0}^{\infty }E_{n}(x) \frac{t ^{n}}{n!}=\frac{2}{ e^{t}+1} e^{xt}. $$

(1.4)

The two-variable special polynomials from application viewpoint are very important as they allow the descent of a bunch of handy, advantageous and pragmatic identities in a fairly simple way. They also prove to be handy in originating new clan of special polynomials. The two-variable families of the Appell polynomials were originated by Bretti *et al.* [3] with the usage of an iterated isomorphism. The two-variable truncated-exponential, Hermite, Legendre and Laguerre polynomials along their extensions are investigated and examined in [2, 5–7, 23] by several authors.

The properties of the truncated-exponential polynomials (TEP) are comparatively little known, despite the fact that these polynomials prove to be very handy in solving many problems of quantum mechanics and optics. The main definition of TEP [1] is given as follows:

$$ e_{n}(x)=\sum_{k=0}^{n} \frac{x^{k}}{k!}. $$

(1.5)

It is noteworthy here that

$$ \lim_{n\rightarrow \infty }e_{n}(x)=e^{x}. $$

The comprehensive investigation and examination for the first time of certain properties of \(e_{n}(x)\) was made by Dattoli *et al.* [6].

The most remarkable properties of these polynomials can be established by using (1.5). An integral representation of these polynomials is given by

$$ e_{n}(x)=\frac{1}{n!} \int _{0}^{\infty }e^{-\xi }(x+\xi )^{n} \,\mathrm{d}\xi , $$

(1.6)

which is a notable consequence of the following well-flourished expression [1]:

$$ n!= \int _{0}^{\infty }e^{-\xi } \xi ^{n} \,\mathrm{d}\xi . $$

(1.7)

The TEP can also be written in terms of the ordinary generating function as follows [6]:

$$ \sum_{n=0}^{\infty }e_{n}(x)t^{n}= \frac{e^{xt}}{1-t}\quad \bigl(t\in \mathbb{C}; \vert t \vert < 1 \bigr). $$

(1.8)

A further extension of the TEP \(e_{n}(x)\) to two variables was given by Dattoli *et al.* [6]. The TEP has shown to play a vital and key role in evaluating integrals containing products of special functions. They also emerge in numerous problems of quantum mechanics and optics, but their properties are not known in a way they should be.

Recalling that the two-variable TEP \(e_{n}(x,y)\) are determined by means of the generating relation (see [6])

$$ \sum_{n=0}^{\infty }{}_{[2]}e_{n}(x,y)t^{n}= \frac{e^{xt}}{1-yt^{2}} $$

(1.9)

and possess the following series definition:

$$ {}_{[ 2]}e_{n}(x,y)=\sum_{k=0}^{[\frac{n}{2}]} \frac{y^{k} x^{n-2k}}{ (n-2k)!}. $$

(1.10)

Recalling also that the higher-order two-variable TEP \(e_{n}(x,y)\) are determined by the generating relation given by (see [6])

$$ \sum_{n=0}^{\infty }{}_{[s]}e_{n}(x,y)t^{n}= \frac{e^{xt}}{1-yt^{s}}, $$

(1.11)

which satisfy the following formula:

$$ {}_{[s]}e_{n}(x,y)=\sum_{k=0}^{[\frac{n}{s}]} \frac{y^{k} x^{n-sk}}{ (n-sk)!}. $$

(1.12)

In view of Eqs. (1.8), (1.9) and (1.11), we find that

$$ {}_{[2]}e_{n}(x,y):=e_{n}^{(2)}(x,y)\quad \text{and}\quad e_{n}(x):=e _{n}^{(1)}(x,1). $$

We note that

$$ U_{n}(y)={}_{[2]}e_{n}(0,y), $$

(1.13)

where \(U_{n}(y)\) represents the Chebyshev polynomials of the second kind, which is determined by the following ordinary generating relation [1]:

$$ \sum_{n=0}^{\infty }U_{n}(x)t^{n}= \frac{1}{1-2xt+t^{2}} \quad \bigl( \vert t \vert < 1;x\leqq 1 \bigr). $$

(1.14)

Furthermore, under the operation of the multiplicative operator \(\widehat{\mathcal{M}}\) and the derivative operator \(\widehat{\mathcal{M}}\), we get

$$ \widehat{\mathcal{M}}_{e^{(s)}}=x+syD_{y}yD_{x}^{s-1} $$

(1.15)

and

$$ \widehat{\mathcal{P}}_{e^{(s)}}=D_{y}, $$

(1.16)

respectively. It follows from (1.15) and (1.16) that the higher-order two-variable TEP \({}_{[s]}e_{n}(x,y)\) are *quasi-monomial* ([24] and [23]).

The idea of the monomiality principle traces back to the year 1941, when Steffenson [25] introduced the concept and method of poweroid. Subsequently, this method was modified by Dattoli [5]. According to the hypothesis of monomiality, the operators \(\widehat{\mathcal{M}}\) and \(\widehat{\mathcal{P}}\) occur and perform as multiplicative and derivative operators for a given polynomial set \(\{q_{n}(x)\}_{n\in \mathbb{N}}\), that is, they satisfy the following relations:

$$ q_{n+1}(x)=\widehat{\mathcal{M}}\bigl\{ q_{n}(x)\bigr\} $$

(1.17)

and

$$ n q_{n-1}(x)=\widehat{\mathcal{P}}\bigl\{ q_{n}(x)\bigr\} . $$

(1.18)

The set \(\{q_{n}(x)\}_{n\in \mathbb{N}}\) operated upon by the multiplicative and derivative operators is then called a quasi-monomial set and must obey the following relation:

$$ {}[ \widehat{\mathcal{P}},\widehat{\mathcal{M}}]= \widehat{\mathcal{P}} \widehat{\mathcal{M}}-\widehat{\mathcal{M}} \widehat{\mathcal{P}}=\widehat{1}, $$

(1.19)

which obviously exhibits a structure of the Weyl group.

If the underlying set \(\{q_{n}(x)\}_{n\in \mathbb{N}}\) is quasi-monomial, its properties can be obtained from those of the operators \(\widehat{\mathcal{M}} \) and \(\widehat{\mathcal{P}}\). Specifically, the following properties hold true:

- (i)
\(q_{n}(x)\) exhibits the differential equation given by

$$ \widehat{\mathcal{M}}\widehat{\mathcal{P}}\bigl\{ q_{n}(x)\bigr\} =n q_{n}(x) $$

(1.20)

if \(\widehat{\mathcal{M}}\) and \(\widehat{\mathcal{P}}\) have differential realizations.

- (ii)
\(q_{n}(x)\) can be explicitly formulated as follows:

$$ q_{n}(x)=\widehat{\mathcal{M}}^{n} \{1\} $$

(1.21)

with the initial condition \(q_{0}(x)=1\).

- (iii)
The exponential generating relation of \(q_{n}(x)\) can be put in the following form:

$$ e^{t\widehat{\mathcal{M}}}\{1\} =\sum_{n=0}^{\infty }q_{n}(x) \frac{t^{n}}{ n!}\quad \bigl( \vert t \vert < \infty \bigr) $$

(1.22)

by using of the identity (1.21) (see, for details, [5, 6] and [23]).

There is ongoing use of the above-mentioned operational methods in such fields of research as classical optics, quantum mechanics and many areas of mathematical physics. Thus, clearly, these methods provide efficient and powerful means of investigation of various families of polynomials.

This article is organized as follows. In Sect. 2, the truncated-exponential based Frobenius–Euler polynomials are introduced and their several interesting properties are obtained. In Sect. 3, summation formulas are established for these types of polynomials. In the last section (Sect. 4), the truncated-exponential based Apostol-type Frobenius–Euler polynomials are introduced and their quasi-monomial properties are derived.