Diamond’s -adic analogue of the classical function has recently been shown to satisfy the integro-differential equation
where is a Volkenborn integral and is the derivative of . We show that this equation characterizes up to a function with everywhere vanishing second derivative. Namely, every solution of (∗) is infinitely differentiable and satisfies .
We show that the set of solutions of the homogeneous equation
associated to (∗) is an infinite-dimensional commutative and associative -adic algebra under the product law
the unit being . We also study Morita’s alternate -adic analogue of and prove similar results.
Cite this article
Eduardo Friedman, On the integro-differential equation satisfied by the -adic -function. Comment. Math. Helv. 85 (2010), no. 3, pp. 535–549DOI 10.4171/CMH/204