Asymptotic distribution of the zeros of recursively defined non-orthogonal polynomials

Let g be a normalized arithmetic function. We define polynomials Qng(x)=x∑k=1ng(k)Qn−kg(x),Q0g(x)≔1.It is known that the case g=id involves Chebyshev of second kind Qnid(x)=xUn−1(x2+1). In this paper we study zero distribution non-orthogonal associated with sn=n2. show zeros Qns(x) are real, simple, and located in (−63,0]. Nn(a,b) number between −63≤a<b≤0. Then determine density function v(x), such limn→∞Nn(a,b)n=∫abv(x)dx.The satisfy four-term recursion. present detail an analysis fundamental roots give answer to open question on recent work by Adams Tran–Zumba. extend method proposed Freud for orthogonal more general systems polynomials. underlying moments distribution.