Electronic Journal of Differential Equations,
Vol. 2011(2011), No. 02, pp. 1-13.

Title: Initial-value problems for first-order differential recurrence 
       equations with auto-convolution

Author: Mircea Cirnu (Univ. Politehnica of Bucharest, Romania)

Abstract:
 A differential recurrence equation consists of a sequence  of
 differential equations, from which must be determined by
 recurrence a sequence of unknown functions. In this article, we
 solve two initial-value problems for some new types of nonlinear
 (quadratic) first order homogeneous differential recurrence
 equations, namely with discrete auto-convolution and  with
 combinatorial auto-convolution of the unknown functions. In
 both problems, all initial values  form a geometric progression,
 but in the second problem the first initial value is exempted 
 and has a prescribed form.
 Some preliminary results showing the importance of the
 initial conditions are obtained by reducing the differential
 recurrence equations to algebraic type.
 Final results about solving the considered initial
 value problems, are shown by mathematical induction. However,
 they can also be shown by changing the unknown
 functions, or by the generating function method.
 So in a remark, we give a proof of the first theorem by the
 generating  function method.


Submitted September 10, 2010. Published January 04, 2011.
Math Subject Classifications: 11B37, 34A12.
Key Words: Differential recurrence equations; discrete auto-convolution;
           algebraic recurrence equations.