Eigenvalues of Hadamard powers of large symmetric Pascal matrices (Q2484227)

Let \(x\) be a real number and let \(A=(a_{ij})\) be nonnegative matrix of order \(n.\) The matrix \(A^{(x)}=(a_{ij}^x)\) of order \(n\) obtained by raising each entry of \(A\) to the power \(x\) is called Hadamard power of \(A.\) The symmetric Pascal matrix of order \(n\) is the matrix \(S_n\) with entries \({i+j-2}\choose{j-1}\) for \(i,j=1,2,\ldots ,n.\) The regularized symmetric Pascal matrix is given by \(R_n= {{2n-2}\choose{n-1}}^{-1} S_n.\) For \(x>0,\) let \(\mu _n(x)\) and \(\tau _n(x),\) respectively, denote the Perron root (maximal eigenvalue) and the trace of the Hadamard power \(R_n^{(x)}.\) The authors prove that \(\liminf _{n\rightarrow \infty }\mu _n(x)\geq \frac{4^x}{4^x-1}\) and \(\lim _{n\rightarrow \infty }\tau _n(x)= \frac{4^x}{4^x-1}\) for all \(x>0.\) When \(x\) is a positive integer say \(k\) then \(\lim _{n\rightarrow \infty }\mu _n(k)= \frac{4^k}{4^k-1}.\) The authors also make two equivalent conjectures related to their results.