arXiv subjects
Publications and source records attributed to Vladimir Kostov.
The {\em spectrum} of the partial theta function $θ:=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ is the set of values of $q\in \mathbb{C}$, $0<|q|<1$, for which $θ(q,.)$ has a multiple zero. We show that the only element of the spectrum belonging to the disk $\mathbb{D}_{0.31}$ is $0.3092493386\ldots$.
The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $θ(q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $θ$. These series are of the form $-q^{-j}+(-1)^jq^{j(j-1)/2}(1+\sum _{k=1}^{\infty}g_{j,k}q^k)$. The coefficients of the stabilized series are expressed by the positive integers $r_k$ giving the number of partitions into parts of three different kinds. They satisfy the recurrence relation $r_k=\sum _{ν=1}^{\infty}(-1)^{ν-1}(2ν+1)r_{k-ν(ν+1)/2}$. Set $(H_{m,j})~:~(\sum _{k=0}^{\infty}r_kq^k) (1-q^{j+1}+q^{2j+3}-\cdots +(-1)^{m-1}q^{(m-1)j+m(m-1)/2})= \sum _{k=0}^{\infty}\tilde{r}_{k;m,j}q^k$. Then for $k\leq (m+2j)(m+1)/2-1-j$ and $j\geq (2m-1+\sqrt{8m^2+1})/2$ one has $g_{j,k}=\tilde{r}_{k;m,j}$.
In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}.
Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ (resp. $A_j\in c_j$) satisfying the equality $M_1... M_{p+1}=I$ (resp. $A_1+... +A_{p+1}=0$)}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. We give new examples of existence of such $(p+1)$-tuples of matrices $M_j$ (resp. $A_j$) which are {\em rigid}, i.e. unique up to conjugacy once the classes $C_j$ (resp. $c_j$) are fixed. For rigid representations the sum of the dimensions of the classes $C_j$ (resp. $c_j$) equals $2n^2-2$.