An asymptotic expansion of the \(p\)-adic Green function
The Green function \[ G(x)=\int_{\mathbb{Q}_p^n}\frac{\chi _p((k,x))}{| (k,k)| _p+m^2} dk,\quad x\in \mathbb{Q}_p^n,\;m>0, \] where \(\chi_p\) is an additive character on \(\mathbb{Q}_p\), was introduced by Vladimirov and Volovich as a \(p\)-adic counterpart of the Green function appearing in Euclidean quantum field theory. The asymptotics of \(G(x)\) for large \(| x| _p\) was calculated by Vladimirov for \(n=1\) and Bikulov for \(n=2\). The author finds the asymptotics for any even \(n\). This and related problems were also studied by the reviewer for a situation where a general quadratic form \(Q(k)\) was taken instead of \((k,k)\), but it was assumed that \(Q(k)\neq 0\) for \(k\neq 0\). The latter is possible only for \(n\leq 4\) [See \textit{A. N. Kochubei}, Theor. Math. Phys. 96, 854-865 (1993); translation from Teor. Mat. Fiz. 96, 123-139 (1993; Zbl 0843.35147); Proc. Steklov Inst. Math. 203, 105-113 (1995; Zbl 0904.11037); Izv. Ross. Akad. Nauk., Ser. Mat. 62, No. 6, 103-124 (1998); English translation to appear in Izv., Math.].
scientific article; zbMATH DE number 1230534