. Soit-?-q,-i-n, a pas de circuit ; soit ? q,i n'est pas à boucle préfixe ; soit ? q,i est à boucle préfixe mais l'étiquette de l'unique transition entrante dans l'état initial de l'automate sous-jacent d'entrée de ? est différente de l'entrée des transitions

P. Angrand, S. Lombardy, and J. Sakarovitch, On the number of broken derived terms of a rational expression, J. Automata, Languages and Combinatorics, vol.151, issue.2, pp.27-51, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00620323

P. Angrand and J. Sakarovitch, Radix enumeration of rational languages, RAIRO - Theoretical Informatics and Applications, vol.44, issue.1, pp.19-36, 2010.
DOI : 10.1051/ita/2010003
URL : https://hal.archives-ouvertes.fr/hal-00647052

P. Angrand and J. Sakarovitch, On the enumerating series of an abstract numeration system, p.5711, 1108.
URL : https://hal.archives-ouvertes.fr/hal-00617617

V. Antimirov, Partial derivatives of regular expressions and finite automaton constructions, Theoretical Computer Science, vol.155, issue.2, pp.291-319, 1996.
DOI : 10.1016/0304-3975(95)00182-4

M. Béal, O. Carton, C. Prieur, and J. Sakarovitch, Squaring transducers: an efficient procedure for deciding functionality and sequentiality, Theoretical Computer Science, vol.292, issue.1, pp.45-63, 2003.
DOI : 10.1016/S0304-3975(01)00214-6

J. Berstel, Transductions and Context-Free Languages, Teubner, 1979.

J. Berstel and C. Reutenauer, Les séries rationnelles et leurs langages Translation : Rational Series and Their Languages, 1984.

V. Berthé and M. Rigo, Combinatorics, Automata and Number Theory, 2010.
DOI : 10.1017/CBO9780511777653

A. Brüggemann-klein, Regular expressions into finite automata, Theoretical Computer Science, vol.120, issue.2, pp.197-213, 1993.
DOI : 10.1016/0304-3975(93)90287-4

J. A. Brzozowski, Derivatives of Regular Expressions, Journal of the ACM, vol.11, issue.4, pp.481-494, 1964.
DOI : 10.1145/321239.321249

P. Caron and M. Flouret, Glushkov Construction for Multiplicities, Proc. CIAA Lecture Notes in Comput, 2000.
DOI : 10.1007/3-540-44674-5_5

P. Caron and D. Ziadi, Characterization of Glushkov automata, Theoretical Computer Science, vol.233, issue.1-2, pp.1-2, 2000.
DOI : 10.1016/S0304-3975(97)00296-X

J. Champarnaud, É. Laugerotte, F. Ouardi, and D. Ziadi, From regular weighted expressions to finite automata, Int. J. Found. Comput. Sci, vol.15, pp.5-687, 2004.

J. Champarnaud, F. Ouardi, and D. Ziadi, An Efficient Computation of the Equation K-automaton of a Regular K-expression, Fundam, Inform, vol.90, pp.1-2, 2009.

J. Champarnaud and D. Ziadi, Computing the Equation Automaton of a Regular Expression in O(s 2) Space and Time, Proc. CPM Lecture Notes in Comput. Sci. n ? 2089, pp.157-168, 2001.
DOI : 10.1007/3-540-48194-X_15

J. Champarnaud and D. Ziadi, FROM C-CONTINUATIONS TO NEW QUADRATIC ALGORITHMS FOR AUTOMATON SYNTHESIS, International Journal of Algebra and Computation, vol.11, issue.06, pp.707-736, 2001.
DOI : 10.1142/S0218196701000772

J. Champarnaud and D. Ziadi, Canonical derivatives, partial derivatives and finite automaton constructions, Theoretical Computer Science, vol.289, issue.1, pp.137-163, 2002.
DOI : 10.1016/S0304-3975(01)00267-5

C. Choffrut, Une caracterisation des fonctions sequentielles et des fonctions sous-sequentielles en tant que relations rationnelles, Theoretical Computer Science, vol.5, issue.3, pp.325-337, 1977.
DOI : 10.1016/0304-3975(77)90049-4

C. Choffrut and M. Goldwurm, Rational transductions and complexity of counting problems, Mathematical Systems Theory, vol.8, issue.4, pp.437-450, 1995.
DOI : 10.1007/BF01185866

C. Choffrut and M. P. Schützenberger, Decomposition de Fonctions Rationnelles, Proc. STACS Lecture Notes in Comput. Sci. n ? 210, pp.213-226, 1986.
DOI : 10.1007/3-540-16078-7_78

C. Frougny, F. Representations, and I. E. , Fibonacci representations and finite automata, IEEE Transactions on Information Theory, vol.37, issue.2, pp.393-399, 1991.
DOI : 10.1109/18.75263

C. Frougny, Number representation and finite automata, Math. Systems Theory, vol.25, pp.37-60, 1992.
DOI : 10.1017/CBO9780511777653.003
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.158.2072

C. Frougny, On the Sequentiality of the Successor Function, Information and Computation, vol.139, issue.1, pp.17-38, 1997.
DOI : 10.1006/inco.1997.2650

V. P. Glushkov, THE ABSTRACT THEORY OF AUTOMATA, Russian Mathematical Surveys, vol.16, issue.5, pp.1-53, 1961.
DOI : 10.1070/RM1961v016n05ABEH004112

A. Khorsi, F. Ouardi, and D. Ziadi, Fast equation automaton computation, Journal of Discrete Algorithms, vol.6, issue.3, pp.433-448, 2008.
DOI : 10.1016/j.jda.2007.10.003

P. B. Lecomte and M. Rigo, Numeration Systems on a Regular Language, Theory of Computing Systems, vol.34, issue.1, pp.27-44, 2001.
DOI : 10.1007/s002240010014

S. Lombardy and J. Sakarovitch, Derivation of Rational Expressions with Multiplicity, Theoret. Computer Sci, vol.332, pp.141-177, 2005.
DOI : 10.1007/3-540-45687-2_39
URL : https://hal.archives-ouvertes.fr/hal-00620039

S. Lombardy, J. Sakarovitch, and . Theor, How expressions can code for automata, RAIRO - Theoretical Informatics and Applications, vol.39, issue.1, pp.217-237, 2005.
DOI : 10.1051/ita:2005013
URL : https://hal.archives-ouvertes.fr/hal-00016385

S. Lombardy and J. Sakarovitch, Corrigendum to our paper : How expressions can code for automata, RAIRO -Theor, Inf. and Applic, vol.443, pp.339-361, 2010.

R. Mcnaughton and H. Yamada, Regular Expressions and State Graphs for Automata, IEEE Transactions on Electronic Computers, vol.9, issue.1, pp.39-47, 1960.
DOI : 10.1109/TEC.1960.5221603

M. Nivat, Transductions des langages de Chomsky, Annales de l???institut Fourier, vol.18, issue.1, pp.339-455, 1968.
DOI : 10.5802/aif.287

R. Paige and R. E. Tarjan, Three Partition Refinement Algorithms, SIAM Journal on Computing, vol.16, issue.6, pp.973-989, 1987.
DOI : 10.1137/0216062

D. Perrin, Finite Automata, pp.1-53, 1990.
DOI : 10.1016/B978-0-444-88074-1.50006-8
URL : https://hal.archives-ouvertes.fr/hal-00620817

C. Reutenauer, Une caractérisation de la finitude de l'ensemble des coefficients d'une série rationnelle en plusieurs variables non commutatives, C. R. Acad. Sci. Paris, vol.284, issue.1877, pp.1159-1162

M. Rigo, Numeration systems on a regular language: arithmetic operations, recognizability and formal power series, Theoretical Computer Science, vol.269, issue.1-2, pp.1-2, 2001.
DOI : 10.1016/S0304-3975(01)00184-0

M. Rigo, Numeration Systems: A Link between Number Theory and Formal Language Theory, Proc. DLT 2010 Lecture Notes in Comput. Sci. n ? 6224, pp.33-53, 2010.
DOI : 10.1007/978-3-642-14455-4_6

J. M. Rutten, Behavioural differential equations: a coinductive calculus of streams, automata, and power series, Theoretical Computer Science, vol.308, issue.1-3, pp.1-53, 2003.
DOI : 10.1016/S0304-3975(02)00895-2

J. Sakarovitch, Eléments de théorie des automates, Vuibert English translation : Elements of Automata Theory, 2003.

M. P. Schützenberger, Sur les relations rationnelles, Proc. Automata Theory and Formal Languages Lecture Notes in Comput. Sci. n ? 33, pp.209-213, 1975.
DOI : 10.1007/3-540-07407-4_22

M. P. Schützenberger, Sur une variante des fonctions sequentielles, Theoretical Computer Science, vol.4, issue.1, pp.47-57, 1977.
DOI : 10.1016/0304-3975(77)90055-X

J. Shallit, Numeration Systems, Linear Recurrences, and Regular Sets, Information and Computation, vol.113, issue.2, pp.331-347, 1994.
DOI : 10.1006/inco.1994.1076

K. Thompson, Regular expression search algorithm, Comm. Assoc. Comput. Mach, vol.11, pp.419-422, 1968.

A. Weber and R. Klemm, Economy of Description for Single-Valued Transducers, Information and Computation, vol.118, issue.2, pp.327-340, 1995.
DOI : 10.1006/inco.1995.1071

D. Wood, Theory of Computation, 1987.

S. Yu, Regular Languages, pp.41-111, 1997.
DOI : 10.1007/978-3-642-59136-5_2

D. Ziadi, J. Ponty, and J. Champarnaud, Passage d'une expression rationnelle à un automate fini non-déterministe, Bull, Belg. Soc. Math, pp.177-203, 1997.