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\rightline{KA-TP-11-2001} \psn
\rightline{June 2001}
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\centerline{ \bf Riccati meets Fibonacci}\psn
\centerline{Wolfdieter L a n g}\psn 
\centerline{\it {Institut f\"ur Theoretische Physik}}\psn  
\centerline{\it{ Universit\"at Karlsruhe} }\pn
\centerline{\it {Kaiserstrasse 12, D-76128 Karlsruhe,}}\pn
\centerline{\it{Germany}}
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The generating functions for generalized {\sl Fibonacci} and {\sl Lucas} 
numbers are shown to be the unique solutions to {\sl Riccati} equations 
with certain initial conditions. This observation leads directly to recursion 
relations for $k$-fold convolutions of these numbers which have been partly 
found earlier in a different way. 
A generalization of this method to so called $p-${\sl Fibonacci} numbers is 
also considered.     
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AMS MSC numbers: 11B83, 11B39, 11C08 
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$^{*}$ E-mail: wolfdieter.lang@physik.uni-karlsruhe.de, \pn 
http://www-itp.physik.uni-karlsruhe.de/${\ \tilde{}}\ $wl 
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