Index transforms with the squares of Bessel functions

New index transforms, involving the squares of Bessel functions of the first kind as the kernel, are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third-order partial differential equation, involving the Laplacian, is obtained

Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values

Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli and Euler numbers and the values of Riemann's zeta function (s). To do this, we explore properties of some Sheffer's sequences of polynomials related to the Kontorovich-Lebedev transform. © 2015 by De Gruyter

On some properties of the Abel-Goncharov polynomials and the Casas-Alvero problem

We derive new properties of the Abel-Goncharov interpolation polynomials, relating them to investigate necessary and sufficient conditions for an arbitrary polynomial of degree n to be trivial, i.e. to have the form a(z - b)(n). These results are associated with an open problem, conjectured in 2001 by E. Casas- Alvero. It says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. In particular, we establish determinantal representation of the Abel-Goncharov interpolation polynomials, having its own interest. Among other results are new Sz.-Nagy-type identities for complex roots and a generalization of the Schoenberg conjectured analog of Rolle's theorem for polynomials with real and complex coefficients