where P is a real even polynomial with positive leading coefficient, which is called a potential. The boundary condition is equivalent to y ∈ L(R) in this case. It is well-known that the spectrum is discrete, and all eigenvalues λ are real and simple, see, for example [3, 14]. The spectrum can be arranged in an increasing sequence λ0 < λ1 < . . .. Eigenfunctions y are real entire functions of o...

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of second quotients Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a with non-decreasing $Q$ to belong Laguerre--Polya class type I. also estimate possible number nonreal zeros $Q.$

Let ( ) ( ) ( ) ( ) be a sequence of mutually independent, identically distributed random variables following semi-stable distribution with characteristic function ( ( ) ) . In this work, we obtain the lower bound of the number of real zeros of the random algebraic equation∑ ( ) . ( ) denote the number of real roots must ( ⁄ ) , except for a set of measure at most .

Let I = 〈g1, . . . , gn〉 be a zero-dimensional ideal of R[x1, . . . , xn] such that its associated set G of polynomial equations gi(x) = 0 for all i = 1, . . . , n is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in √ I. We also provide a necessary and sufficient (numerical) condition for all the zeros of G to be in a given se...

Exceptional orthogonal polynomials were introduced by Gomez-Ullate, Kamran and Milson as polynomial eigenfunctions of second order differential equations with the remarkable property that some degrees are missing, i.e., there is not a polynomial for every degree. However, they do constitute a complete orthogonal system with respect to a weight function that is typically a rational modification ...

Rankin and Swinnerton-Dyer [R, S-D] prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A := {eiθ : π 2 ≤ θ ≤ 2π 3 }. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function” in...