Eigenproblem for Jacobi matrices: hypergeometric series solution

Eigenproblem for Jacobi matrices: hypergeometric series solution.

We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The (small) expansion parameters are the entries of the two diagonals of length d-1 sandwiching the principal diagonal that gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3d...

On an inverse eigenproblem for Jacobi matrices

Recently Xu 13] proposed a new algorithm for computing a Jacobi matrix of order 2n with a given n n leading principal submatrix and with 2n prescribed eigenvalues that satisfy certain conditions. We compare this algorithm to a scheme proposed by Boley and Golub 2], and discuss a generalization that allows the conditions on the prescribed eigenvalues to be relaxed.

Mock Jacobi Forms in Basic Hypergeometric Series

We show that some q-series such as universal mock theta functions are linear sums of theta quotient and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are multiplied by suitable powers of q. And we prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or com...

Two modified Jacobi methods for M-matrices

two modified jacobi methods for m-matrices