Coulson-type integral formulas for the sum of powers of absolute values of roots of polynomials

Abstract

Coulson integral formula was initiated in the research of energy of graphs in 1940 by Coulson, which provides an efficient way to calculate the energy of graphs without knowing any eigenvalues of graphs. Recently, Qiao et al. generalized the concept of energy from graphs to polynomials, which is referred to as the general energy of polynomials. For a polynomial ϕ(z) and a nonzero real number α, the general energy of polynomial ϕ(z), denoted by Eα(ϕ), is defined as the sum of αth powers of absolute values of all (nonzero) roots of ϕ(z), i.e., Eα(ϕ)=∑zk≠0|zk|α, where zk represents the nonzero roots of ϕ(z). In a series of three publications by Qiao et al., they gave some beautiful Coulson-type integral formulas for Eα(ϕ) when the roots of ϕ(z) are all nonnegative numbers for any rational number α and any irrational number α with 0<|α|<1, and when the roots of ϕ(z) are all real numbers for any rational number α and any irrational number α with 0<|α|<2. For the remaining α, their integral formulas are invalid. In this paper, we set up a research procedure covering all the nonzero real numbers α. That is to say, it not only combines all the Coulson-type integral formulas established by Qiao et al., but also involves in all the remaining unsolved α. As an extension and generalization of known integral formulas, the Coulson-type integral formulas for Eα(ϕ), when the roots of ϕ(z) are all pure imaginary numbers, are also presented, for any nonzero real number α. Moreover, we also give an alternative form of integral formulas expressed by the coefficients of polynomials.