Explicit formulas for Euler polynomials and Bernoulli numbers

In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

Some Symmetric Properties and Location Conjecture of Approximate Roots for (p,q)-Cosine Euler Polynomials

In this paper, we introduce (p,q)-cosine Euler polynomials. From these polynomials, we find several properties and identities. Moreover, we find the circle equations of approximate roots for (p,q)-cosine Euler polynomials by using a computer.

Hankel determinants of sequences related to Bernoulli and Euler polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.