Let {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number real zeros, say

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

let$ p_{n}(x)= sum_{i=0}^{n} a_{i}x^{i}$ be a random algebraicpolynomial, where $a_{0},a_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. thus $a_j$&apos;s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}h_{j}(t)}$, where $h_j(t)$&apos;s are complex slowlyvarying functions.under the assumption that th...

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...