Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions (Q2039328)

The authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\dots-2\cos(\theta_d)-z}d\theta.\] Such investigation was mainly restricted to dimensions \(d=1,2\). In the Introduction, they start to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \( \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To get rid of the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G(\frac{-2d\epsilon}{1-\epsilon},n)\), for values of \(\epsilon\in (0,1]\). In this way, they succeed in representing \(G(z,n)\) as a convergent series (see, e.g., Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past. As a whole, this paper is complementary to the thors' previous paper [J. Funct. Anal. 277, No. 4, 965--993 (2019; Zbl 1496.47011)] in which the authors have shown that \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\dots,n\), the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) -- the singular part of \(G(z,n)\) -- was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\), \(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix~A.