Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but largely forgotten chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-exi...

The purpose of this note is to give a combinatorial proof of the three-term linear recurrence for Motzkin numbers. The present work is inspired by R emy’s combinatorial proof of the linear recurrence for Catalan numbers (RAIRO Inform. Theor. 19(2) (1985) 179) and the more recent proof given by Foata and Zeilberger (J. Combin. Theory Ser. A 80(2) (1997) 380) for Schr; oder numbers (Z. Math. Phys...

We prove that the sequence (eSn )n≥0 of excursions in the quarter plane corresponding to a nonsingular step set S ⊆ {0,±1} with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. Moreover, we display th...

This paper studies global asymptotic stability for positive solutions to the equation yn = yn−k + yn−m 1 + yn−k yn−m , n = 0, 1, . . . , with y−m , y−m+1, . . . , y−1 ∈ (0,∞) and 1 ≤ k < m. The paper includes a discussion of stability for a wide class of symmetric rational difference equations which includes the type studied here as well as several other in the recent literature. c © 2006 Elsev...

In this note, we consider the following rational difference equation: xn+1 = f (xn−r1 , . . . , xn−rk )g(xn−m1 , . . . , xn−ml )+ 1 f (xn−r1 , . . . , xn−rk )+ g(xn−m1 , . . . , xn−ml ) , n= 0,1, . . . , where f ∈ C((0,+∞)k, (0,+∞)) and g ∈ C((0,+∞)l, (0,+∞)) with k, l ∈ {1,2, . . .}, 0 r1 < · · ·< rk and 0 m1 < · · ·<ml , and the initial values are positive real numbers. We give sufficient con...