Basic hypergeometric polynomials with zeros on the unit circle

Polynomials with Symmetric Zeros

“…Other results on the distribution of zeros of SI polynomials include the following: Sinclair & Vaaler [49] showed that a monic SI polynomial p(z) of degree n satisfying the inequalities L r p(z) 2+2 r (n − 1) 1−r or L r p(z) 2+2 r (l − 2) 1−r , where r 1, L r p(z) = |p 0 | r +· · ·+|p n | r and l is the number of non-null terms of p(z), has all their zeros on S; the authors also studied the geometry of SI polynomials whose zeros are all on S. Choo & Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on S were considered in [51,52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials.…”

“…and l is the number of non-null terms of p(z), has all their zeros on S; the authors also studied the geometry of SI polynomials whose zeros are all on S. Choo & Kim applied Theorem 11 to SI polynomials in [50]. Hypergeometric polynomials with all their zeros on S were considered in [51,52]. Kim [53] also obtained SI polynomials which are related to Jacobi polynomials.…”

Dragging the roots of a polynomial to the unit circle

“…The linear scaling suggests that maybe the interlace number should be replaced by a normalized version, dividing it by another function of p that scales linearly. Indeed, Theorem 4.6 suggests the use of 1 2 ||p|| 1 as normalizing function, which would restrict the interlace number to the interval (0,1]. At this point, a normalization looks more like a nuisance than an advantage (introducing annoying denominators when dealing with integer polynomials, for instance), so we will keep the interlace number as is.…”

“…For real rooted polynomials there are many classical families of orthogonal polynomials and several polynomials arising in Combinatorics, either as counting functions or as generating functions (Chudnovsky and Seymour [11],Brändén [5], Savage and Visontai [34], Brändén et al [6]). For circle rooted polynomials, such families are somewhat less common (see Dilcher and Robins [14], Lalín and Smyth [27], Lalín and Rogers [26], Area et al [1], Botta, Marques, and Meneguette [4], Lee and Yang [28]).…”

“…We can look at p(x) as a perturbation of x n + 1, and consider, for real ε, the polynomial x n + 1 + εp(x), asking how small ε has to be for angle-interlacing and circle rootedness to set in. It turns out to be more convenient to scale the parametric polynomial by α = 1/ε, and define, for any complex polynomial p of degree at most n, the family (1) p α (x) = α(x n + 1) + p(x),…”