The main aim of this paper is the development of suitable bases (replacing the power basis x (n ∈ N≥0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms...

We study linear combinations of exponentials en, λn ∈ Λ in the case where the distance between some points λn tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L(0, T ). Here we prove tha...

Linear combinations of exponentials e iλ k t in the case where the distance between some points λ k tends to zero are studied. D. Ull-rich [30] has proved the basis property of the divided differences of exponentials in the case when {λ k } = Λ (n) and the groups Λ (n) consist of equal number of points all of them are close enough to n, n ∈ Z. We have generalized this result for groups with arb...

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester’s formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determi...