On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

Discrete time heat kernel and UV modified propagators with dimensional deconstruction

“…where we should notice that the sum starts from t = 1. We follow the similar path as the derivation of f c ν (z) in [38]. We find that the series sum f c ν (z) satisfies…”

“…The discrete time heat kernels for the D-dimensional canonical scalar model was discussed in [19]. The discrete time heat kernels for the graph Laplacian are recently dealt with in [38].…”

“…and that the differential equation (2.5) is recovered by the continuum limit, since 1 ϵ ∆ → ∂ ∂s , if ϵ → 0. The solution of (3.3) can be expressed by using the discrete modified Bessel function I c ν (t) [38,44,45],…”

“…Now, let us return to our present DD model. For the heat kernel from the forward difference equation, the simplest modification method described above turns out not to be effective, because I c 0 (0) = 1, I c n (t) = 0 for t < n [38] and a = 1 − ϵf 2 (2 + p 2 /f 2 ) → −∞ when p 2 /f 2 → ∞, it is not immediately clear if the elimination of finite terms for small numbers t makes the propagator behaves better at high energy (p 2 /f 2 → ∞). Another idea for the heat kernel from the forward difference equation, we consider eliminating the infinite terms of the even powers of a in the sum over t. Indeed, the behavior of G(p 2 ) at large p 2 becomes better, but the sign change of a at large p 2 results in a 'cut' in the complex plane of p 2 that are difficult to interpret.…”

“…Therefore, it is significant to study the mathematical properties of the discrete time heat kernel, albeit for a simple model. Fortunately, discrete time heat kernel for a certain class of graph Laplacians has recently been discussed in [38], so we can manage to apply it to our calculations.…”