# American Institute of Mathematical Sciences

2005, 2005(Special): 730-737. doi: 10.3934/proc.2005.2005.730

## Unique summing of formal power series solutions to advanced and delayed differential equations

 1 Department of Mathematics, East Carolina University, Greenville, NC 27858, United States, United States

Received  September 2004 Revised  May 2005 Published  September 2005

The analytic delayed-differential equation $z^2 \psi ^{\ \! \prime } (z) \ + \ \psi (z/q) \ = \ z$ for $q>1$ has a solution which can be expressed as a formal power series. A $q$-advanced Laplace-Borel kernel provides for the construction of an analytic solution whose domain is the right half plane with vertex at the initial point $z=0$. This method is extended to provide a continuous family of solutions, of which a subfamily extends to a punctured neighborhood of $z=0$ on the logarithmic Riemann surface. Conditions are given on the asymptotics of $\psi ^{\ \! \prime } (z)$ near $z=0$ to ensure uniqueness.
Citation: David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730
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