Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.

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ABSTRACT. Let Q n (a;;/3,7,c) be the polynomials of degree n which satisfy the recurrence relation: a!n+cQn+i(a;/3,7,c) + a! n+c _iQ n _i(z;/3,7,c) + (Pn+c + {3Sn,o)Qn(x; /?,7,c) = x(l + (7 -l)Sn,0)Qn(a; P, 7> c), Q_l(a;;/3,7,c) = 0, Qo(a;;^,7,c) = 1.In the above, P is real, 7 > 0, a n + c and /3n+c are real sequences with Q: n + C > 0, and 5 ni o is the Kronecker symbol. These polynomials are called scaled co-recursive associated polynomials. The co-recursive associated orthogonal polynomials are obtained from the above for 7=1. In this paper, the Newton sum rules for the k-th power of the zeros of scaled co-recursive associated orthogonal polynomials are determined in terms of the Newton sum rules of associated orthogonal polynomials. Some monotonicity properties of the zeros also are given.

We prove that for ν > n − 1 all zeros of the nth derivative of Bessel function of the first kind Jν are real. Moreover, we show that the positive zeros of the nth and (n + 1)th derivative of Bessel function of the first kind Jν are interlacing when ν ≥ n, and n is a natural number or zero. Our methods include the Weierstrassian representation of the nth derivative, properties of the Laguerre-Pólya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivative of the Struve function of the first kind Hν are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel and Struve functions of the first kind. Some conjectures and open problems related to Hurwitz theorem on the zeros of Bessel functions are also proposed, which may be of interest for further research.

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