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On generalized zeta functions of formal languages and series

1991
*
Discrete Applied Mathematics
*

.,

doi:10.1016/0166-218x(91)90097-g
fatcat:2k7nh4ov6nds7bhc5yrsnddz2u
*On**generalized**zeta**functions**of**formal**languages**and**series*, Discrete Applied Mathematics 32 (1991) 141-153. We study*generalized**zeta**functions**of**formal**languages**and**series*. ... We show that it is decidable whether or not the (*generalized*)*zeta**function**of*a Q-algebraic*series*is a rational*function*. ... Acknowledgement The author would like to thank Professor Arto Salomaa for his useful comments*on*the preliminary version*of*this paper. ...##
###
On algebraic generalized zeta functions of formal power series

1991
*
Theoretical Computer Science
*

Honkala, J , 0n algebraic

doi:10.1016/0304-3975(91)90156-v
fatcat:rjlosugn55bwvp5wusbaadpo7m
*generalized**zeta**functions**of**formal*power*series*, Theoretical Computer Science 79 (1991) 263-273. We study algebraic*generalized**zeta**functions**of**formal*power*series*. ... We show that the*generalized**zeta**function**of*a rational*series*is an algebraic*function*if*and*only if it is a root*of*a rational*function*. ... Introduction The*zeta**functions**and**generalized**zeta**functions**of**formal**languages**and*power*series*were defined by Berstel*and*Reutenauer [2] . ...##
###
Zeta functions of recognizable languages
[chapter]

1988
*
Lecture Notes in Computer Science
*

Motivated by symbolic dynamics

doi:10.1007/3-540-19488-6_109
fatcat:wdvlerfplveu7jijhbnlcyogoe
*and*algebraic geometry over finite fields, we define cyclic*languages**and*the*zeta**function**of*a*language*. ... The main result is that the*zeta**function**of*a cyclic*language*which is recognizable by a finite automaton is rational. ... Introduction Motivated by algebraic geometry over finite fields*and*symbolic dynamics, we call*zeta**function**of*a*formal**language*L the*function*t n (L) = exp (Y. a n ) n where a n is the number*of*words ...##
###
Page 439 of Mathematical Reviews Vol. , Issue 92a
[page]

1992
*
Mathematical Reviews
*

Summary: “We study algebraic

*generalized**zeta**functions**of**formal*power*series*. ... Edmund Robinson (Brighton) 92a:68071 68Q45 Honkala, Juha (SF-TURK)*On*algebraic*generalized**zeta**functions**of**formal*power*series*. ...##
###
Zeta Functions of Formal Languages

1990
*
Transactions of the American Mathematical Society
*

Motivated by symbolic dynamics

doi:10.2307/2001573
fatcat:xaq4dlbi2fasfku5e2pclwojs4
*and*algebraic geometry over finite fields, we define cyclic*languages**and*the*zeta**function**of*a*language*. ... The main result is that the*zeta**function**of*a cyclic*language*which is recognizable by a finite automation is rational. ...*One*may suppose that G corresponds, in the above isomorphism, to the set G' = {U, g, A)lg E G}*and*that x = (j, h, 11). As x 2 =J= 0,*one*has P llj =J= O. Similarly, P).j =J= O. ...##
###
Algebraicity of the zeta function associated to a matrix over a free group algebra
[article]

2013
*
arXiv
*
pre-print

Following

arXiv:1303.3481v4
fatcat:hgc2nrk6tjdldcshv3wb3ix6om
*and**generalizing*a construction by Kontsevich, we associate a*zeta**function*to any matrix with entries in a ring*of*noncommutative Laurent polynomials with integer coefficients. ... We show that such a*zeta**function*is an algebraic*function*. ... Acknowledgement We are grateful to Yves André, Jean-Benoît Bost*and*Carlo Gasbarri for their help in the proof*of*Theorem 4.4. ...##
###
Zeta functions of formal languages

1990
*
Transactions of the American Mathematical Society
*

Motivated by symbolic dynamics

doi:10.1090/s0002-9947-1990-0998123-x
fatcat:gbd67mksezbktmkunperindelm
*and*algebraic geometry over finite fields, we define cyclic*languages**and*the*zeta**function**of*a*language*. ... The main result is that the*zeta**function**of*a cyclic*language*which is recognizable by a finite automation is rational. ...*One*may suppose that G corresponds, in the above isomorphism, to the set G' = {U, g, A)lg E G}*and*that x = (j, h, 11). As x 2 =J= 0,*one*has P llj =J= O. Similarly, P).j =J= O. ...##
###
Algebraicity of the zeta function associated to a matrix over a free group algebra

2014
*
Algebra & Number Theory
*

Algebraicity

doi:10.2140/ant.2014.8.497
fatcat:lc5egz4a4rhd7iy4gm6xptuqna
*of*the*zeta**function*associated to a matrix over a free group algebra Christian Kassel*and*Christophe Reutenauer Following*and**generalizing*a construction by Kontsevich, we associate a*zeta*... We show that such a*zeta**function*is an algebraic*function*. ... We are also indebted to François Bergeron*and*Pierre Guillot for assisting us with computer computations in the process detailed in Remark 5.2, to Frédéric Chapoton for his comments*on*the last part*of*...##
###
Page 3369 of Mathematical Reviews Vol. , Issue 91F
[page]

1991
*
Mathematical Reviews
*

The

*generalized**zeta**function**of*a*language*L is Z(L), where L is the characteristic*series**of*L. By definition, a*language*... They define the*generalized**zeta**function*Z(S)*of*a*formal*power*series*S in noncommuting variables by Z(S) = exp(>>,,>; 7(Sn)/m), where 2(S,,) is the homogeneous part*of*degree n*of*S viewed in a canonical ...##
###
N-algebraicity of zeta functions of sofic-Dyck shifts
[article]

2017
*
arXiv
*
pre-print

As a consequence the

arXiv:1501.05843v2
fatcat:vmi2t53ah5hbjdzrshygfcwnfq
*zeta**function**of*a sofic-Dyck shift is the*generating**function**of*a visibly pushdown*language**and*is thus an N-algebraic*series*. ... We prove that the multivariate*zeta**function**of*a sofic-Dyck shift is the commutative*series**of*some visibly pushdown*language*. ... The*zeta**function**of*a sofic-Dyck shift is the*generating**series**of*some computable visibly pushdown*language*. It is thus N-algebraic. Proof. ...##
###
Cyclic languages and strongly cyclic languages
[chapter]

1996
*
Lecture Notes in Computer Science
*

The result is used to give another proof

doi:10.1007/3-540-60922-9_5
fatcat:ciwexlcprrfrffbxwhc7s5qeja
*of*the rationality*of*the*zeta**function**of*rational cyclic*languages*. ... We prove that cyclic*languages*are the boolean closure*of**languages*called strongly cyclic*languages*. ...*Zeta**function**of*a cyclic*language*We rst give the de nitions*of**generalized**zeta**function**and**zeta**function**of*a*language**of*nite words over a nite alphabet A. ...##
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Page 5135 of Mathematical Reviews Vol. , Issue 92i
[page]

1992
*
Mathematical Reviews
*

Adam (H-AOS)
92i:68111 68Q70 05A15 68Q45 68Q50
Honkala, Juha (SF-TURK)

*On**generalized**zeta**functions**of**formal**languages**and**series*. Discrete Appl. Math. 32 (1991), no. 2, 141-153. ... The*generalized**zeta**function*Z(r)*of*a noncommutative*formal*power*series*r over an alphabet X with coefficients in a subsemiring A*of*the field*of*real numbers is Z(r) = exp()/,,>;(1/m)c(rn)), where ...##
###
Rationality of the Möbius function of subword order

1992
*
Theoretical Computer Science
*

We prove the rationality

doi:10.1016/0304-3975(92)90379-t
fatcat:izjan5fvovff3gf6sxwxpgtmgq
*of*various noncommutative*formal*power*series*, whose coefficients are determined by the Mobius*function*or*zeta**function**of*the subword partial order*of*noncommutative monomials ... Bjiirner, A.*and*C. Reutenauer, Rationality*of*the Mobius*function**of*subword order, Theoretical Computer Science 98 (1992) 53-63. ... We give here examples*of*such*series*. We show that the*zeta**function**and*the Mobius*function**of*subword order,*and*their powers, when viewed as*formal**series**of*words, are rational. ...##
###
p-adic model theory, p-adic integrals, Euler products, and zeta functions of groups
[article]

2020
*
arXiv
*
pre-print

We then briefly discuss

arXiv:2007.09242v1
fatcat:nolcfxmrhjh4tnsuwpghi5p43e
*zeta**functions*arising from definable equivalence relations*and*p-adic elimination*of*imginaries, which have applications to counting representations*of*groups. ... We give a survey*of*Denef's rationality theorem*on*p-adic integrals, its uniform in p versions, the relevant model theory,*and*a number*of*applications to counting subgroups*of*finitely*generated*nilpotent ... While Igusa had used the rationality*of*his local*zeta**functions*to prove a conjecture*of*Borevich*and*Shafarevich*on*rationality*of*a Poincare*series*counting points*on*a variety modulo powers*of*p, Denef ...##
###
On a zeta function associated with automata and codes

2007
*
Theoretical Computer Science
*

The

doi:10.1016/j.tcs.2007.04.004
fatcat:dqrgwdisbvfrjmx6xlyagtzgsa
*zeta**function**of*a finite automaton A is exp{ ∞ n=1 a n z n n }, where a n is the number*of*bi-infinite paths in A labelled by a bi-infinite word*of*period n. ... It reflects the properties*of*A: aperiodicity, nil-simplicity, existence*of*a zero. The results are applied to codes. ... Although*zeta**functions**of**languages**and*sofic systems have been previously considered, this*zeta**function*seems to be new: it may not be obtained by associating with the automaton some*language**and*then ...
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