# Vietoris type theorem related to positivity of trigonometric polynomials

## TL;DR

This paper establishes a Vietoris type theorem for the positivity of sine and cosine sums with specific sequences, leading to new insights on zeros of trigonometric polynomials and positive sums for orthogonal polynomials.

## Contribution

It introduces a novel Vietoris type theorem for trigonometric sums, improving zero location estimates and deriving new positive sums for orthogonal polynomials.

## Key findings

- Positivity conditions for sine and cosine sums are established.
- Improved bounds for zeros of certain trigonometric polynomials.
- New positive sums for classes of orthogonal polynomials.

## Abstract

In this work, a Vietoris type theorem for the positivity of sine and cosine sum for a particular sequence of real numbers is provided. In this connection, the positivity of a particular type of sine sum involving ratio of some parameters is given, which is new in the literature. Various new results that follow from the Vietoris type theorem include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for orthogonal polynomials. An open problem is also provided for the partial sums of the generalized polylogarithm.