Pairs of lattice paths and positive trigonometric sums
نویسندگان
چکیده
Ismail et al. (Constr. Approx. 15 (1999) 69–81) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.
منابع مشابه
Lattice Paths and Positive Trigonometric Sums
A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coeecients is given. The proof uses lattice paths, and identiies the trigonometric sum as a polynomial with positive integer coeecients. Some special cases of the q-analog conjectured by Bressoud are established, and new conjectures are given.
متن کاملLattice path enumeration and Toeplitz matrices
This paper is about counting lattice paths. Examples are the paths counted by Catalan, Motzkin or Schröder numbers. We identify lattice paths with walks on some path-like graph. The entries of the nth power of the adjacency matrix are the number of paths of length nwith prescribed start and end position. The adjacency matrices turn out to be Toeplitz matrices. Explicit expressions for eigenvalu...
متن کاملConvergence, Uniqueness, and Summability of Multiple Trigonometric Series
In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let E be a subset of positive (Lebesgue) measure of the k dimensional torus. The principal result is that the convergence of a trigonometric series on E forces the boundedness of the partial sums alm...
متن کاملPositive trigonometric sums and applications
Some new positive trigonometric sums that sharpen Vietoris’s classical inequalities are presented. These sharp inequalities have remarkable applications in geometric function theory. In particular, we obtain information for the partial sums of certain analytic functions that correspond to starlike functions in the unit disk. We also survey some earlier results with additional remarks and comments.
متن کاملA Few Finite Trigonometric Sums
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interest...
متن کامل