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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A pair of difference differential equations of Euler-Cauchy type


Author: David M. Bradley
Journal: Trans. Amer. Math. Soc. 355 (2003), 4985-5002
MSC (2000): Primary 34K06; Secondary 34K12, 34K25
DOI: https://doi.org/10.1090/S0002-9947-03-03223-9
Published electronically: July 24, 2003
MathSciNet review: 1997592
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Abstract: We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.


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Additional Information

David M. Bradley
Affiliation: Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, Maine 04469-5752
Email: dbradley@member.ams.org, bradley@math.umaine.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03223-9
Keywords: Difference differential equations, integral transforms, adjoint relation, Dickman-de Bruijn function, sieves
Received by editor(s): March 19, 2002
Received by editor(s) in revised form: October 15, 2002
Published electronically: July 24, 2003
Additional Notes: This research was supported by the University of Maine summer faculty research fund.
Article copyright: © Copyright 2003 American Mathematical Society