Related papers: On polynomials of binomial type, Ramanujan-Soldner…
We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers--Ramanujan continued fraction play central roles in…
In this note, we consider applications of Ratner's theorem to constructions of families of polynomials with dense values on the set of primitive integer points from the viewpoint of invariant theory.
A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal…
We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on…
We study the properties of the logarithm of the derivative operator and show that its action on a constant is not zero, but yields the sum of the logarithmic function and the Euler-Mascheroni constant. We discuss more general aspects…
We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials, while the second one treats operators linear…
An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results…
Some integral properties of Jack polynomials, hypergeometric functions and invariant polynomials are studied for real normed division algebras.
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…
The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as…
In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…
We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete…
For operators generated by a certain class of infinite band matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order finite difference equations. This…
We introduce a new class of polynomials $\{P_{n}\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic,…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field…