We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a β-density function gR , m: [0 , ∞) ⟶ R, where (R, m) is the homogeneous coordinate ring associated with the toric pair (X, D), such that ∫0∞gR,m(x)dx=β(R,m),where β(R, m) is the second coefficient of the Hilbert–Kunz function for (R, m) , as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function gR , m: [0 , ∞) ⟶ R is compactly supported and is continuous except at finitely many points, (2) the function gR , m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk–Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.