The author introduces an automorphism of the space of continuous functions from the \(p\)-adic integers to the complex \(p\)-adic numbers, i.e.,the completion of the algebraic closure of the field of \(p\)-adic numbers. The space of locally analytic functions is invariant under this automorphism. By applying the inverse of this automorphism to the locally analytic function \(x\mapsto r^x\), where \(r\) is an integer congruent \(1\) modulo \(p\), he is able to recover the \(p\)-adic incomplete Gamma function by A. O’Desky and D. H. Richman [Trans. Am. Math. Soc. 376, No. 2, 1065–1087 (2023; Zbl 1523.33009)]. Further, this automorphism is seen to be self-adjoint with respect to a non-degenerate symmetric bilinear form on the function space. This bilinear form is defined using \(p\)-adic integration and convolution of the corresponding Mahler series with respect to a certain measure on the \(p\)-adic integers.
From an explicit action of integer powers of the automorphism, he is able to construct an integral transform in terms of convolutions and convolution powers with \(p\)-adic integers. Thus, he obtains an expression for the \(p\)-adic incomplete Gamma function in terms of this new integral transform. Finally, he observes that this integral transform takes continuous functions \(\mathbb{Z}_p\to \mathbb{C}_p\) to locally analytic functions and gives an explicit representation of its image in terms of the automorphism, and also proves a convolution theorem involving Mahler coefficients. As an application, the differential equation \(F'+F=G\) with \(G\in \mathbb{C}_p[[t]]\) having bounded coefficients is studied.
The automorphism \(S\) itself is defined as: \[ S(\phi)(x)=\phi(x)-x\phi(x-1) \] for \(\phi:\mathbb{Z}_p\to\mathbb{C}_p\) a continuous function.