The functional-differential equation is closely related to Golomb’s self-described sequence , We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number there is exactly one increasing solution with as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact we conjecture that the difference of two increasing solutions behaves very similarly as the error term in the asymptotic expression (where is the golden number).
L’équation différentielle fonctionnelle a des liens étroits avec la suite auto-décrite de Golomb, Nous décrivons les solutions croissantes de cette équation. Nous montrons qu’une telle solution possède nécessairement un point fixe non négatif, et que pour chaque nombre il y a exactement une solution croissante ayant pour point fixe. Nous montrons également qu’en général une condition initiale ne détermine pas une solution unique: les courbes représentatives de deux solutions croissantes distinctes se croisent en effet une infinité de fois. En fait, nous conjecturons que la différence de deux solutions croissantes se comporte de façon très similaire au terme d’erreur dans l’expression asymptotique (où est le nombre d’or).
@article{JTNB_1999__11_1_211_0, author = {Y.-F. S. P\'etermann and J.-L. R\'emy and I. Vardi}, title = {On a functional-differential equation related to {Golomb's} self-described sequence}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {211--230}, publisher = {Universit\'e Bordeaux I}, volume = {11}, number = {1}, year = {1999}, doi = {10.5802/jtnb.247}, zbl = {0973.11029}, mrnumber = {1730441}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.247/} }
TY - JOUR TI - On a functional-differential equation related to Golomb's self-described sequence JO - Journal de Théorie des Nombres de Bordeaux PY - 1999 DA - 1999/// SP - 211 EP - 230 VL - 11 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.247/ UR - https://zbmath.org/?q=an%3A0973.11029 UR - https://www.ams.org/mathscinet-getitem?mr=1730441 UR - https://doi.org/10.5802/jtnb.247 DO - 10.5802/jtnb.247 LA - en ID - JTNB_1999__11_1_211_0 ER -
Y.-F. S. Pétermann; J.-L. Rémy; I. Vardi. On a functional-differential equation related to Golomb's self-described sequence. Journal de Théorie des Nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 211-230. doi : 10.5802/jtnb.247. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.247/
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