Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.6

On Arithmetic Partial Differential Equations

Pentti Haukkanen and Jorma K. Merikoski
School of Information Sciences
FI-33014 University of Tampere

Timo Tossavainen
School of Applied Educational Science and Teacher Education
University of Eastern Finland
P. O. Box 86
FI-57101 Savonlinna


Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic partial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equations, we give a complete solution, and for others, we extend previously known results. For example, we determine under which conditions two consecutive partial derivations are commutative.

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(Concerned with sequences A000040 A003415.)

Received March 21 2016; revised versions received September 12 2016; September 14 2016. Published in Journal of Integer Sequences, October 11 2016.

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