Gaussian integration formulas for logarithmic weights and application to 2-dimensional solid-state lattices

The making of Gaussian numerical integration formulas is considered for weight functions with logarithmic singularities. Chebyshev modified moments are found most convenient here. The asymptotic behavior of the relevant recurrence coefficients is stated in two conjectures. The relation with the recursion method in solid-state physics is summarized, and more details are given for some two-dimensional lattices (square lattice and hexagonal (graphene) lattice).