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math.NT2008★ 1 cited
Arithmetic differential operators on Z_p
A. Buium, C. C. Ralph, S. R. Simanca
We prove that a function f from Z_p to itself is analytic if and only if it can be represented as f(x)=F(x, dx, ..., d^r x) where dx=(x-x^p)/p is the Fermat quotient operator and F…
math.NT2008
Arithmetic Laplacians
Alexandru Buium, Santiago R. Simanca
We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives a…
math.NT2008★ 3 cited
Arithmetic partial differential equations, II: modular curves
Alexandru Buium, Santiago R. Simanca
We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstei…