Rational convolution roots of isobaric polynomials

Convolution polynomials

The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically.

Multinomial convolution polynomials

In 9] Knuth shows how to derive the convolution formulas of Ha-gen, Rothe and Abel from Vandermonde's convolution or binomial theorem for integer exponents. In the present paper, we shall rst present a short and elementary proof of the multi-extension of the above con-volution formulas, due to Raney and Mohanty. In the second part we shall present a multi-version of Knuth's approach to convolut...

Differential Operators and Weighted Isobaric Polynomials

where m ∈ Z. We are interested in these operators as linear operators on a special ring of polynomials, discussed in [5], namely, the ring of isobaric polynomials Λ̃, a ring isomorphic to the ring of symmetric functions Λ. The polynomials in Λ̃, or more precisely Λ̃k, are over indeterminants t1, . . . , tk and the isomorphism just mentioned is given by identifying tj with the signed elementary sym...

Roots of Polynomials

A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product an(x − r1)(x − r2) · · · (x − rn), where an is the leading coefficient and r1, r2, . . . , rn are all of its n complex roots. We will look at how to find roots, or zeros, of polynomials in one variable. The solution of multi-variate polynomials can often be transformed ...

Polynomials , roots , and interlacing