ARKIV FOR MATEMATIK Band 2 nr 22. By CHRISTER LECH

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1 ARKIV FOR MATEMATIK Band 2 nr 22 Counicated 26 Noveber 952 by F. CARLSON A note on recurring series By CHRISTER LECH We wish to prove the following theore, which extends to any field of characteristic 0 a result, proved by SKOLE~ [4] in the field of rational nubers and by MAttLER [2] in the field of algebraic nubers. Theore. In a /ield o/ characteristic 0, let a sequence c~ v=0,,2,... satis/y a recursion ]orula o/ the type Cv~O~lC~,-l+~2Cv-2+"" + O~nCv-n v=n,n+l,n+2,... I/ c~ = 0 /or in/initely any values o/ v, then those c~ that are equal to zero occur periodically in the sequence /ro a certain index on. It will be shown by an exaple that the restriction for the characteristic is essential (section 6). Fro the theore can be deduced a characterization of those sequences {c~} that contain 0 (or: any nuber) an infinity of ties (see MAHLER [2]). In particular, only a finite nuber of the c~ can be equal to zero if the quotient of two different roots of the equation l=alt+~2t2+"'+ant ~ is never a root of unity. SKOLEM and MAHLER used for their proofs a p-adic ethod, due to SKOLEM [3]. Our proof will closely follow that of MAHLER, and is partly built on it.. A sequence {c~} in any field ay be considered as the TAYLOR-coefficients of a rational function if it satisfies a. linear recursion forula as above. By resolving this rational function into partial fractions it is possible to get an explicit expression for the c~. In a field of characteristic 0 we get cx = Y~ Af Pj (x) x = 0,, 2..., where the Pj (x) are polynoials whose coefficients, together with the Aj, are algebraic over the field that is generated by the c~. Therefore, to prove the theore, it is sufficient to prove the following lea. Lea. Let the /unction F (x) be defined by F (x) = 5 Ay Pj (x), = 47

2 C. LECH, 2t note on recurring series where the Pj (x) are polynoials whose coe//icients, together with the Aj, belong to a /ield K o/ characteristic O. I~ F (x) = 0/or an in/inity o/rational integral values o/ x, then there is a natural nuber r and di//erent nubers r a), r a)... r (~) o/ the set O,..., r-, so that all the values r(l~ +rz, ra) +rz,..., r(~ +rz z=0, l,+_2,... and, in addition, F (x) vanish. at ost a finite nuber o] rational integral values o] x ake Without loss of generality we can suppose that K is just the field generated by the Aj and the coefficients of the Pj (x). Denote by P the field of rational nubers. 2. First assue that K is algebraic over P. This is the case treated by MArlLER. We give a sketch of his proof. Choose a prie ideal p of K, dividing neither nuerator nor denoinator of any Aj. Denote by [,tlp (cock) the corresponding valuation of K, nored so that I PIp=, where p is the natural P prie divided by p. (In the sequel this special noring of a valuation will be tacitly assued.) Applying a generalization of FERMAT'S theore we can find a natural nuber r, so that () IA~-Ilo < v-; ~ =, 2,...,. As a consequence of these inequalities, the functions A~ x j=l, 2...,, where x is restricted to rational integral values, can be represented by p-adic power series of x. Instead of F (x) consider now the r functions Fo(z)=.F(a+rz)= ~ATA~zPj(a+rz) (r=0,,..., r-. = For rational integral values of z these functions can all be represented by p- adic power series. If any of the has an infinity of rational integral zeros, it vanishes identically. In fact, the zeros then have a p-adic point of accuulation in the region of convergence of the power series, and, as in the case of a power series of a coplex variable, this is ipossible unless the series vanishes identically. Fro the connection between F (x) and the r functions F, (z) we get the desired result about the rational integral zeros of F (x). 3. Now assue that K is not algebraic over its prie field P. Observe that if we were able to find a valuation of K, continuating a p-adic valuation of P, so that a precise analogue of the inequalities () could be proved, then the reasoning of the preceding section could be carried through. Actually we shall construct such a valuation of K. Denoting it by ~ (:r (~ e K) and eaning by ~ the corresponding natural prie = ~ we have to prove that for ~oe natural nuber r 48

3 ARKIV FOR MATEMATIK~ Bd 2 nr 22 (2) 9(AI-) <p,- ]=, Since K is generated by a finite nuber of eleents, it ust be of finite transcendence-degree k over P. It ay therefore be considered as an algebraic extension of a field P(~I..., ue), where gi... gk are algebraically independent over P. This algebraic extension ust be finite and hence siple, as the characteristic is 0. Thus every eleent of K ay be written in the for (~) Q(ul..., zd teo(zi,/..., xk) +Pl (ul... zk) P~-I (Zl..., ~k) ~l-)~ where ), satisfies an equation, irreducible in P (~,---, z~), ~9(~... ~k'~)~-~l JvCl (gl..., ~k)~ l- "~- "'" ~- Cl(gl, o.., ~k) = O, Q(Ul,..., uk), the P,(Ul... gk) and the C,(ul,..., uk)being polynoials with rational integral coefficients. Let Aj (~... uk ; ~) (i =, 2... ) be expressions of the for (3) for the nubers Aj and denote by D (zl..., uk) the discriinant of? (Zl..., ~; 2), regarded as a polynoial in 2. After these preliinaries we proceed to the construction of the valuation. 5. Choose k rational integers a..., ak, so that when these nubers are substituted for ~... ~, then D (ul..., x~) and the denoinators of the Aj (ul... uk;2) do not vanish. Fix in P[2] one irreducible factor of the polynoial ~(a..., ak; 4) and let v q be a root of this factor, thus ~0 (a~..., ak;v q)=0. According to section 2 there are a valuation of the field P (v~), corresponding to soe prie ideal p, and a natural nuber r, such that (4) I(Ai(ai,..., a~; vq))r-~<p-~ =-~ i=, 2,...,, p denoting the natural prie divided by p (]p],=l). We denote by P (v ~) the perfect p-adic extension of P (v~).- Let e~,..., ~ be eleents of P(v ~) which are algebraically independent over P). We have k J[~/ lira t (a. + p~ e.) - a~ ], = 0 v =, 2... k, which eans that in the sense of the valuation of P(v~ i the nubers a~+p~el,...,a~+p~e~ tend to al,...,a~ respectively when n tends to infinity. We assert that for large values of n there is also in P (v~) a root 2 (") ~) The existenco of such eleents e~,..., e~: follows fro the fact that P (v ~) has the cardinality of the continuu. To obtain an effective construction, let p, (x~..., xk) (v=, 2, 3... ) be an enueration of all polynoials in k variables with rational integral coefficients. Write el... ek as infinite sus of rational nubers. Define these sus by steps, every step consisting in the addition of one ter to each su and the rth step having no influence on ]p~(e~... ek) p (p<v) but assuring that p~ (e~,..., ek)% 0. This construction can be precisod by soe univocal law. 49

4 C. LECH, A note on recurring series of the equation ~ (a + pn el... ak + pn ek ; x) = 0,. such that A (n) tends to v a when n tends to infinity. For it is clear that the polynoial congruence (al+pnsl,...,ak+pnek;x)--~v(al,...,ak;x) (odp n) is satisfied. Furtherore, for large values of n the discriinant of q0 (al +pn el,..., ak +pn ek; x) will have the sae valuation as the nuber D (al,..., ak), which is different Iro zero. According to HENSEL ([], pp , we can therefore deduce, since there is a factorization in P (v a) (a... a~ ; x) = (x - 0) ~p (x), that, for large n, there is a factorization in P (tg) with (a + io n sj..., ak + p~ ek ; x) = (x - 2 (~)) ~on (x) li 2(~) - v~ Ip = 0. n=oo As the rational functions in a valued field are continuous in the sense of the valuation, we can find an n for which (5) ] (At (al+pn el '.-., ak +p~ ek; 2(n))) ~-(Aj (al,..., ak; v~)) ~ Ip <p p Fixing this value of n, we have, by (4) and (5), =,2,...,. (6) I(Aj(al+pnel,...,ak+pn~:;2(n)))~--ll~<p ~,- i=,2.... The valuation of K = P (zl, 9 9 x~ ; 2) is now obtained by an ebedding of this field in the valued field P (zg) in confority to the forulas ~t~ -+ a, + p n e, (~). It is indeed evident fro our choice of the e, and 2 (n) that the field P (a q- + P" el..., ak + p" ek ; 2 (n)) ( ~ P (va)) is isoorphic to K. The validity of the inequalities (2) is expressed by (6). Our proof is thus copleted. 6. There are fields of prie characteristic in which the stateent of the theore (and of the lea) does not hold. We shall show this by an exaple. Let e be the unity eleent of the prie field with p eleents and let u be an indeterinate over the sae field. The sequence satisfies the recursion forula c~=(e+g)~-e-u" v=0,, 2... c, = (2 e + 2,) c,_ - (e + 3 z + z ~) c,-2 + (~ + ~2) c~-3. It is seen that c~=0 if and only if v is an integral power of p. 420

5 ARKIV FOR MATEMATIK. Bd 2 nr 22 REFERENCES [] K. HENSEL, Theorie der algebraischen Zahlen I, Leipzig und Berlin, 908. [2] K. MAHLER, Eine arithetisehe Eigenschaft der Taylor-koeffizienten ralionaler Funktionen, Akad. Wetensch. Asterda, Proc. 38, (935). [3] TH. SKOLEM, Einige S~tze tiber gewisse Reihenentwicklungen und exponenti~le Beziehungen it Anwendung auf diophantische Gleichungen. Oslo Vid. akad. Skrifter I 933 Nr. 6. [4] ----, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, C. r. 8 congr, scand. ~ Stockhol 934, Tryckt den 6 februari 953 Uppsala 953. A_lqvist & Wiksells Boktryckeri AB 42

Bernoulli Numbers. Junior Number Theory Seminar University of Texas at Austin September 6th, 2005 Matilde N. Lalín. m 1 ( ) m + 1 k. B m.

Bernoulli Numbers. Junior Number Theory Seminar University of Texas at Austin September 6th, 2005 Matilde N. Lalín. m 1 ( ) m + 1 k. B m. Bernoulli Nubers Junior Nuber Theory Seinar University of Texas at Austin Septeber 6th, 5 Matilde N. Lalín I will ostly follow []. Definition and soe identities Definition 1 Bernoulli nubers are defined