2023/04/29 by Jay Jorgenson, Anders Karlsson, Jorgenson, Jay +3 · 2 citations
Mathematics · #05C99 #11L03 #35K08 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematical functions and polynomials #Number Theory (math.NT) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2305.00202
openalex publication_date 2023/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Xm denote the discrete circle with m vertices. For x,y∈ Xm and complex s, let GXm,χβ(x,y;s) be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on Xm that are twisted by a character χβ. We will compute GXm,χβ(x,y;s) in two different ways. First, using the spectral expansion of the Laplacian, we show that GXm,χβ(x,y;s) is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing β or s appropriately, the sums in question involve powers of the secant function. Second, by viewing Xm as a quotient space of ℤ, we prove that GXm,χβ(x,y;s) is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of GXm,χβ(x,y;s), these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the L-functions associated to the cycle graph Xm, thus answering a question from arXiv:2212.13687v1.