Generalized Power Sums Research Articles

Generalized Power Sum and Newton-Girard Identities

In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard power sum identity. © 2020, Springer Japan KK, part of Springer Nature.

A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.

On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums

Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.

Identities of Symmetry for Generalized Euler Polynomials

We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the -adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.

Point lattices and oscillating recurrence sequences†

We consider the following question: which real sequences that satisfy a linear recurrence with constant coefficients are positive for sufficiently large n? We show that the answer is negative for both and if the dominating characteristic roots in the representation of as a generalized power sum comprise either two pairs of conjugate complex roots or one negative real root and one pair of conjugate complex roots. The proof uses results from Diophantine approximation and the geometry of numbers. Concerning the case of a real positive dominating root we show what the answer to the question of positivity is for almost all values of the other dominating roots, provided that all dominating roots are simple.

Representations and traces of the Hecke algebras H n(q) of type A n−1

The notion of the connectivity class of minimal words in the algebra Hn(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented. Guided by these results, the connection between the Ocneanu trace on Hn(q) and Schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of Hn(q). They appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions. Their evaluation involves the use of the Littlewood–Richardson rule, which is proved to apply to Hn(q) just as it does to Sn. Both representation matrices and characters are tabulated.

Characters of Hecke algebras Hn(q) of type An-1

The connection between the Ocneanu trace on Hn(q) and Schur functions leads to a simple method for calculating the irreducible characters of the Hecke algebras Hn(q). The characters appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions.

Optimal weighted chebyshev-type quadrature formulas

A weighted quadrature formula is called of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes. Among such quadrature rules we construct an optimal one, i. e., one which has maximum algebraic degree of accuracy and minimum error when applied to the first power not exactly integrated. Optimal quadrature rules, typically, have multiple nodes. Their construction requires the complete solution of systems of algebraic equations involving generalized power sums. Numerical results are presented for the case of constant weight function on a finite interval, as well as for weight functions of the Hermite and Laguerre type on infinite intervals.

Generalized power sums

Generalized power sums