Journal of Number Theory. On Diophantine approximations of Ramanujan type q-continued fractions

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1 Journal of Number Theory ) Contents lists aailable at ScienceDirect Journal of Number Theory On Diophantine approximations of Ramanujan type q-continued fractions Tapani Matala-aho Ville Merilä Matematiikan laitos Oulun Yliopisto Linnanmaa PL Oulun Yliopisto Finland article info abstract Article history: Receied 24 October 2007 Reised 20 June 2008 Aailableonline24February2009 Communicated by Michael A. Bennett MSC: 11J82 11A55 11B65 Keywords: Irrationality measure q-continued fraction q-series We shall consider arithmetical properties of the q-continued fractions K q sn S 0 + S 1 q n + + S h q hn ) T 0 + T 1 q n + +T l q ln S i T i q K q < 1 and some related continued fractions where is a fixed aluation of an algebraic number field K and s hl N. Inparticularweget sharp irrationality measures for certain Ramanujan Ramanujan Selberg Eisenstein and Tasoe continued fractions Elseier Inc. All rights resered. 1. Introduction There are a few works considering arithmetical properties of q-continued fractions. In general the research is concentrated on studying irrationality measures of the Rogers Ramanujan continued fraction RRq t) = 1 + qt q 2 t q 3 t in archimedean imaginary quadratic fields see Bundschuh [6] Osgood [17] Shiokawa [18] and Stihl [19]. Matala-aho [12] considered approximations also in other aluations and proed some higher degree quantitatie irrationality results. For example from [12] one gets * Corresponding author. addresses: tma@cc.oulu.fi T. Matala-aho) merila@paju.oulu.fi V. Merilä) X/$ see front matter 2009 Elseier Inc. All rights resered. doi: /j.jnt

2 T. Matala-aho V. Merilä / Journal of Number Theory ) RR 5 1)/2 ) t / Q 5) RR1/5 t) = 1 + t t t t t / Q m) for any m Q and t Q. Further if p 17 is a prime number and t Q then the p-adic number RRp t) / Q m) m Z. Moreoer RRp t) p 2) aswellasrr1/5 t) hae an effectie irrationality measure μ = 2. By an effectie irrationality measure exponent) of a gien number α C p we mean a numberμ = μα) 2 which satisfies the condition: for eery ɛ > 0 there exists an effectiely computable constant H 0 ɛ) 1 such that α M N > 1 p H μ+ɛ for eery M/N Q with H = max{ M N } H 0 ɛ). In Amou and Matala-aho [1] the Thue Siegel method was used to proe approximation measures for a restricted class of continued fractions K tq n ) s S 0 + S 1 q n t + + S h q hn t h ) T 0 + T 1 q n t + +T l q ln t l S i T i K q < 1. 1) For example in [1] an irrationality measure μ 2 95 was proed for the Watson Ramanujan continued fraction K q n + q 2n q = 1/d d Z \{0 ±1}. 1 In this paper we shall consider q-continued fractions 1) in a more general setting and some related q-fractions too. For example for the both q-fractions K q 2n 1 + q q 2n 1 n K 1 + q n 2) we get the irrationality measure μ = 2ifq = 1/d d Z \{0 ±1}. Note that concerning the latter continued fraction in 2) a deep claim 1 q q q 1 + q 2 q q 3 = q2 ; q 3 ) q; q 3 ) made by Ramanujan is proed in [3] and again quite recently in [4] and [5]. We may tackle also the following Ramanujan Selberg continued fractions S 1 q) = 1 q q + q 2 q 3 q 2 + q = q2 ; q 2 ) q; q 2 3) ) S 2 q) = 1 q + q 2 q 4 q 3 + q 6 q 8 q 5 + q 10 q = q; q8 ) q 7 ; q 8 ) q 3 ; q 8 ) q 5 ; q 8 4) ) S 3 q) = 1 q + q 2 q 2 + q 4 q 3 + q = q; q2 ) q 3 ; q 6 ) 3 5)

3 1046 T. Matala-aho V. Merilä / Journal of Number Theory ) considered in [20] and Eisenstein continued fractions E 1 q) = 1 q q 3 q q 5 q 7 q 3 q 9 q 11 q = q n2 6) n=0 E 2 q) = 1 q q 3 q 5 q q + 1 q q q 7 + = 1) n q n2 7) n=0 deelopments of partial theta series see [7]. At the best we obtain irrationality measures μs 1 1/d)) = μs 2 1/d)) = μe 1 1/d)) = 3ifd Z \{0 ±1}. Next we consider the following Tasoe s continued fractions T 1 u a) = ua + a 2 + ua 3 + a 4 + ua + 5 8) T 2 u a b) = ua + b + ua 2 + b 2 + ua ) Ealuations of numerous ariants of T 1 and T 2 are considered in [8 11]. Later we will show how we may ealuate T 1 and T 2 simply by using the well-known identities from the theory of q- continued fractions such as 27) of the Rogers Ramanujan continued fraction. Consequently we then get μt 1 u a)) = μt 2 u a b)) = 2 where u Q and a b Z \{0 ±1}. Finally we shall study certain continued fractions W = W 1 + W 2 + W 3 + where the partial denominators satisfy a second-order recurrence. Let F n ) F 0 = 0 F 1 = 1denotethe Fibonacci sequence. If we set then μ 2 F ) Notations F = F 1 + F 3 + F 5 + As usual the q-factorials are defined by a) 0 = a; q) 0 = 1 and a) n = a; q) n = 1 a)1 aq) 1 aq n 1 ) for all n Z +.Foranyp of the set P ={ } {p Z + p is a prime} the notation = will be used for the usual absolute alue of C = C and p for the p-adic aluation of the p-adic field C p the completion of the algebraic closure of Q p normalizedby p p = p 1.Let K be an algebraic number field of degree κ oer Q its place and K the corresponding completion. If the finite place of K lies oer the prime p wewrite p for an infinite place of K we write. Further the notation I is used for an imaginary quadratic field. By using the normalized aluations α = α κ /κ κ =[K : Q ] the Height Hα) of α K is defined by the formula Hα) = α α = max{ 1 α }

4 T. Matala-aho V. Merilä / Journal of Number Theory ) and the Height Hα) of ector α = α 1...α m ) K m is gien by Hα) = α α = { } max 1 αi. i=1...m A characteristic λ = λ q = will also be used in the sequel. By the alue of the continued fraction log Hq) log q q 1 q K in C p wemeanthelimit b 0 + K a n = b 0 + a 1 a 2 b n b 1 + b 2 + a 3 b 3 + A n lim n B n of the conergents A n /B n where A n and B n satisfy the recurrences A n = b n A n 1 + a n A n 2 B n = b n B n 1 + a n B n 2 n 2 with initial alues A 0 = b 0 A 1 = b 0 b 1 + a 1 B 0 = 1 B 1 = b Results By using the polynomials St) = S 0 + S 1 t + + S h t h T t) = T 0 + T 1 t + +T l t l C p [t] we define the following q-continued fraction Gq t) = T t) + K tq n ) s Sq n t) T q n s Z +. 10) t) In our main theorem Theorem 1 we study approximation of the alues of the continued fraction 10) by algebraic numbers from a gien number field K. Henceforth we shall fix Case 1): When B + λa > 0 we set { l A = max 2 s + h } B = s ) B μ = μ 1 = B + λa. 12)

5 1048 T. Matala-aho V. Merilä / Journal of Number Theory ) Case 2): Whereas when B + λa = 0 we set ρη + sρ/2) μ = μ 2 = inf ρη + sρ/2) + λa + Aa 2 + η) 13) where a = 2 η + ρ η = s + 1)η3 6s2 η) and the infimum is taken oer all positie parameters 1 < η < 2 and 0 < ρ for which Dρ η s) = ρη + sρ/2) + λ a + Aa 2 + η ) > 0. If λ = 1 and s 1 we hae the rough estimate μ 2 64s s + 21 ) /3 when η = η 0 = 3 + 4s)/2 + 2s) and for ρ we choose ρ = 2ρ 0 where ρ 0 satisfies Dρ 0 η 0 s) = 0. Howeer in the case s = 1 the choice η = ρ = yields μ Theorem 1. Let be a place of K and let q t K satisfy B + λa 0 q < 1 and let St) T t) K[t] satisfy S 0 T 0 0 S 0 1 s 1 and Sq k t)t q k t) 0 for all k N. Then there exist positie constants C i D i and H i i= 1 2 suchthat for any β K with H = max{hβ) H i }. Gq t) β > C i H κμ i/κ +D i log H) 1/2 Obseration 1. Let h = 0 and s 2l > 0 thenb/a = 2 gies an irrationality measure μ 1 = 2/2 + λ) for Gq t) for all q K satisfying 2 <λ 1. Obseration 2. For all t q = b/d Q satisfying d > b 2 the real number Gq t) has an irrationality measure log b/d μ 1 = 2 logb 2 /d) and especially G1/d t) as well as the p-adic number Gp l t) p isaprimel Z + ) hae an irrationality measure μ 1 = 2. WhenK = I is the infinite place of K and q = 1/d d > 1 whered Z K weobtain μ 1 = 2 κ = κ = 2. For more similar explicit calculations see [15]. Corollary 1. a) Let b d Qd> 0 e q I and.if q < 1 and d 2 + 4b is a square of a rational number then ) μ 1 K b = 2 d + eq n 2 + λ. 14)

6 T. Matala-aho V. Merilä / Journal of Number Theory ) b) Let b d Q e q K = Q d 2 + 4b) and.if q < 1 d> 0 d 2 + 4b > 0 and d 2 + 4b isnota square of a rational number then κ = 2 κ = 1 and The continued fraction ) μ 1 K b = 2 d + eq n 2 + λ. 15) K eq n 16) with q = 5 1)/2 belongs to the case b) where λ = 1 μ 1 = 2 and thus 1 K 1 + eq β n > C H4+Dlog H) 1/2 for any β K with H = max{hβ) H 1 }. In particular the alue of continued fraction 16) is not in Q 5). 4. Results from q-series and q-continued fractions Let F t) be a nonzero solution of the q-functional equation qt) s F q 2 t ) = T t)f qt) + St)F t) 17) where s 1 and h l St) = S k t k T t) = T k t k C p [t] k=0 k=0 are of degree h = deg t S l = deg t T. Theorem 2. Let be a place of K S j T j K and let q t K satisfy B + λa 0 q < 1 s 1 T q k t)sq k t) 0 for all k N.LetFt) be a solution of the functional equation 17) such that F q n t ) < c n n N 18) for some positie constant c and the numbers F t) F qt) are not both zero then the two numbers F t) and F qt) are linearly independent oer K and there exist positie constants C i D i and H i such that β1 F t) + β 2 F qt) > C i H κμ i/κ +D i log H) 1/2 for any β = β 1 β 2 ) K 2 \{0} with H = max{hβ) H i }i= 1 2.

7 1050 T. Matala-aho V. Merilä / Journal of Number Theory ) In the case 2) we need to suppose that the solution F t) is analytic see [1]. Here we recall that A = max{l/2s + h)/4}. Proof. The proof follows closely the ideas of [12] and [1] thus we gie only some essential parts of it. First we gie a notation p k t k w qk t k if p k w q k w for all k N used in the sequel. Next we choose an H Q such that T t) w H w 1 + +t l ) St) w H w 1 + +t h ) for all places w. Then by using the iterates of Eq. 17) we get the following approximation formula R n t) = t sn+1) q sn+2 2 ) F q n+2 t ) = T n t)f qt) + S n t)f t) 19) with the upper bounds { max Tn t) w Sn t) } w P w n) = c n w q An 2 w w 20) Rn t) R n) = c n 2 q Bn2 21) where c w = 2s + h + l + 1) ) δ w H w t s+h+l w q 2A w and c 2 = c 1 t s+1 q 2s. Let us proe the estimate 20). Define polynomials T n t) = T q n t ) T n 1 t) + t s q sn S q n t ) T n 2 t) S n t) = T q n t ) S n 1 t) + t s q sn S q n t ) S n 2 t) n Z 1 with initial conditions T 1 t) = 1 S 1 t) = 0 T 0 t) = T t) S 0 t) = St). From Eq. 17) we get t s q sn+1) F q n+2 t ) = T q n t ) F q n+1 t ) + S q n t ) F q n t ) n 0. 22) Multiplying Eq. 22) by term t sn q sn+1 2 ) weobtain t sn+1) q sn+2 2 ) ) ) F q n+2 t = T q n t t sn q sn+1 2 ) ) F q n+1 t + ) t s q sn S q n t t sn 1) q sn 2) ) F q n t ) = T q n t T n 1 t)f qt) + S n 1 t)f t) ) + t s q sn S q n) T n 2 t)f qt) + S n 2 t)f t) ) = T n t)f qt) + S n t)f t).

8 T. Matala-aho V. Merilä / Journal of Number Theory ) Next we note that T q n t ) w q wln H w 1 + +t l ) 23) q sn t s S q n t ) w q w s+h)n H w 1 + +t s+h ). 24) Using the estimates 23) and 24) we hae for X n = S n T n that X n w 2 δ w H w ) { 1 + +t s+h+l max q nl w Xn 1 q s+h)n } w Xn 2 25) where δ w = 0ifp and δ w = 1ifp.Puta 1 = l and a 2 = s + h then a j n + An j) 2 + 2An j) An 2 + 2An j = 1 2 n 2. Thus in iew of 11) the inequality 25) yields X n w 2 δ w H w 1 + +t s+h+l ) q w 2A ) n q w An 2. For the remainder readily holds R n = q sn+2 2 ) t sn+1) since F q n t) c n 1 by assumption. If now ) F q n+2 t t q ) 2 sc n 1 t s q 3s/2 ) n q s 2 n2 c w = 2s + h + l + 1) ) δ w H w t s+h+l w q 2A w and c = q 2s t s+1 c1 we hae the claims 20) and 21). Now we apply Theorems 3.3 and 4.1 from [13] in the case 1). In the case 2) the proof uses a construction based on Thue Siegel s lemma and again the iterates of Eq. 17) see [1]. Note that we obtained an improement to [13] in the dependence on the first parameter of our upper bound term A. Theorem 3. See [14].) Let S 0 T 0 0 S 0 p 1q t C p.ifs 1then T t) + K tq n ) s Sq n t) = St) F t) T q n t) F qt) = St) T 0 Gt) S 0 Gqt) q p < 1 26) where G : C p C p is an analytic function such that F t) = t x Gt) is a solution of the functional equation 17) satisfying q x = S 0 /T 0 and F qt) 0. Moreoerifdeg St) = h = 0 theng: C p C p is an entire function. The conergence in 26) is uniform with respect to ariable t in eery bounded subset of C p. Thus the alue of the q-fraction T 0 + T 1 t + +T l t l + K tq n ) s S 0 + S 1 tq n + + S h t h q hn ) T 0 + T 1 tq n + +T l t l q ln s 1 is a quotient of power series conerging in some disk t p < r R +. As an example we gie the well-known ealuation

9 1052 T. Matala-aho V. Merilä / Journal of Number Theory ) RRq t) = 1 + qt q 2 t q 3 t 1 + = F t) F qt) 27) of the Rogers Ramanujan continued fraction where F t) = 5. Proof of Theorem 1 and the corollaries n=0 q n2 q) n t n q p < 1. Proof of Theorem 1. Let F t) be a continuous nonzero solution of 17) satisfying F 0) 0. First we note that the condition 18) is satisfied by the continuity of F t). Next we show that not both the numbers F t) and F qt) are zero. If on the contrary F qt) = F t) = 0 for some q t C p q p < 1 then F q n t) = 0foralln N by 19). By taking limits we get a contradiction. Then by Theorem 2 the numbers F t) and F qt) are linearly independent oer K and thus both nonzero. Thus by Theorem 2 we get an irrationality measure μ = μ i i = 1 2) for F t)/f qt) which by Theorem 3 implies the truth of our Theorem 1. Proof of Corollary 1. In [16] it is proed that where b c d e q K q < 1 and d + K b + cq n d + eq = α + cq n + eβq 2n 1 n K α + e β)q n 28) α = d + d 2 + 4b 2 β = d d 2 + 4b β < α. 2 Here we note that the formula 28) generalizes the transformation formula 1 + K k + q n 1 q n = α + K α βq n where q C q < 1 k R k > 1/4 α = k)/2 andβ = k)/2 originally considered by Ramanujan for more details see [2]). By setting c = 0 in 28) we obtain d + K b d + eq = α + n K α + e β)q. n eβq 2n 1 In the case a) we assume that d 2 + 4b = m 2 m Q + and d > 0. Thus αβ Q and β < α. Inthe case b) the assumption d > 0implies β < α too. Thus by using the relation 28) we may apply Theorem 1 with the subsequent Obseration 1 in order to get 14) and 15). Proof of the examples. We start by taking an een contraction of the continued fractions 3) 4) and 6). This yields b 0 + K a n = b 0 + b n b 2 b 1 + a 2 + b 2 a 1 a 2 a 3 b 4 b 2 a 4 +b 3 b 4 )+a 3 b 4 +K cn dn 29)

10 T. Matala-aho V. Merilä / Journal of Number Theory ) where c n = a 2n+2 a 2n+3 b 2n+4 b 2n d n = b 2n+2 a 2n+4 + b 2n+3 b 2n+4 ) + a 2n+3 b 2n+4 for each continued fraction. Let us denote the tail d 0 + Kc n /d n ) of 3) by S C 1 q) = 1 + q + q2 + q 3 q q 2 ) q q 3 ) q q 4 ) q 2 + q q) q 3 + q q) q 4 + q q) + whereas S C 2 q) = 1 + q3 + q 4 + q 6 q q 3 ) q q 5 ) q q 7 ) q 5 + q q 2 ) q 7 + q q 2 ) q 9 + q q 2 ) + for the second Ramanujan Selberg continued fraction. Instead for the Eisenstein continued fraction holds d 0 + K c n = E C 1 d q) = 1 + q q3 q 5 n q 8 1 q 4 ) q 14 1 q 6 ) q 20 1 q 8 ) q 3 q q 2 ) q 5 q q 2 ) q 7 q q 2 ) +. The first contraction S C 1 has the parameters s = 3 h = 1 l = 2 and for SC 2 and EC 1 we hae s = 6 h = 2 l = 4. Applying Theorem 1 to these contractions gies the following irrationality measures for the Ramanujan Selberg continued fractions 3) 4) as well as for the Eisenstein fraction 6) i.e. μ 1 Si q) ) = μ 1 E1 q) ) = 3 i = λ The third continued fraction S 3 q) and E 2 q) hae the parameters s = h = 1 l = 0 and s = l = 1 h = 0 denote q 2 by q) respectiely and thus μ 2 S3 q) ) = μ 2 E2 q) ) λ= 1. Next we consider Tasoe s continued fractions T 1 u a) = ua + a 2 + ua 3 + a 4 + ua ) T 2 u a b) = ua + b + ua 2 + b 2 + ua ) First we show how we may ealuate T 1 u a) and T 2 u a a) simply by using the known identity 27) of the Rogers Ramanujan continued fraction. In 30) we set t = 1/au) and q = a 2 which gies directly T 1 u a) = ua) qt q 2 t q 3 t

11 1054 T. Matala-aho V. Merilä / Journal of Number Theory ) Next by 27) we hae T 1 u a) = F qt) uaft) F t) = n=0 q n2 q) n t n q p < 1. Hence we get n=0 T 1 u a) = a n+1)2 u n 1 n /a 2 1) a 2n 1)) a p > 1 32) n=0 a n2 u) n /a 2 1) a 2n 1)) aresultin[8]withu =. Similarly by putting a = b t = 1/au) and q = 1/a in 31) we get T 2 u a a) = n=0 a n+1)n+2)/2 u n 1 n /a 1) a n 1)) n=0 a nn+1)/2 u) n /a 1) a n 33) 1)) aresultin[8]. For the ealuation of T 2 u a b) in the general case we use the known identity see [2]) where 1 + αq βq αq 2 βq = Gαβ) Gαqβ) q p < 1 34) Gαβ)= n=0 α n q nn+1)/2 q) n βq) n. Namely by setting α = u) 1 β = ua) 1 and q = ab) 1 weobtain Hence T 2 u a b) = T 2 u a b) = ua) αq βq αq 2 βq = Gαqβ) uagαβ) q p < 1. n=0 ann+1)/2 b nn 1)/2 / n j=1 a j b j 1)ua j+1 b j + 1) ua n=0 ann+3)/2 b nn+1)/2 / n j=1 a j b j 1)ua j+1 b j + 1) 35) if ab p > 1. Note that in [8] the alue of T 2 u a b) for general a b ab > 1isnotgieninaclosed form but as a quotient of series whose coefficients satisfy certain recurrence relations. Note also that our ealuations 32) 33) and 35) are alid for any a b u C p with a p > 1 ab p > 1 which should be compared to [8 11] where generally only the cases a b Z 2 are considered. Now readily μ 1 T1 u a) ) = λ where λ = λ1/a). After applying the transformation 29) to the continued fraction in 34) we hae also that μ 1 T 2 u a b)) = 2/2 + λ) where λ = λ1/ab).

12 T. Matala-aho V. Merilä / Journal of Number Theory ) Finally consider a continued fraction W = W 1 + W 2 + W 3 + where the denominators are the forms W n = aα n + bβ n αβ = 1 and α β say α > β. Ifwedenoteq = 1/α 2 we get an equialent continued fraction W = K αq n a + bq. n 36) As an example we may take a sequence W n ) defined by the recurrence W n+2 = 3W n+1 W n W 0 = 2 W 1 = 3. If now a b αβ K [K : Q]=2 and α = β thenμ 2 W ) On the other hand let F n ) F 0 = 0 F 1 = 1 denote the Fibonacci sequence. Set α = 1 + 5)/2 a = b = 1/ 5 and q = α 4 thenwehae F = F 1 + F 3 + F 5 + = 1 α q n 1 K a bα 2 q. n 1 Obiously the corresponding continued fraction with een partial denominators F 2n reduces to 36).) Consequently μ 2 F ) too. References [1] M. Amou T. Matala-aho Arithmetical properties of solutions of linear second order q-difference equations in: Number Theoretic Methods Iizuka 2001 in: De. Math. ol. 8 Kluwer Acad. Publ. Dordrecht 2002 pp [2] G.E. Andrews B.C. Berndt Ramanujan s Lost Notebook. Part I Springer New York 2005 xi+437 pp. [3] G.E. Andrews B.C. Berndt L. Jacobsen R.L. Lamphere The continued fractions found in the unorganized portions of Ramanujan s Notebooks Mem. Amer. Math. Soc ) 1992) i+71 pp. [4] G.E. Andrews B.C. Berndt J. Sohn A.J. Yee A. Zaharescu On Ramanujan s continued fraction for q 2 ; q 3 ) /q; q 3 ) Trans. Amer. Math. Soc ) 2003) [5] D. Bowman J. Mclaughlin N.J. Wyshinski A q-continued fraction Int. J. Number Theory 2 4) 2006) [6] P. Bundschuh Ein Satz uber ganze Funktionen und Irrationalitatsaussagen Inent. Math ) [7] A. Folsom Modular forms and Eisenstein s continued fractions J. Number Theory 117 2) 2006) [8] T. Komatsu On Hurwitzian and Tasoe s continued fractions Acta Arith ) 2003) [9] T. Komatsu On Tasoe s continued fractions Math. Proc. Cambridge Philos. Soc ) [10] T. Komatsu Tasoe s continued fractions and Rogers Ramanujan continued fractions J. Number Theory ) [11] T. Komatsu Hurwitz and Tasoe s continued fractions Monatsh. Math ) [12] T. Matala-aho On Diophantine approximations of the Rogers Ramanujan continued fraction J. Number Theory ) [13] T. Matala-aho On Diophantine approximations of the solutions of q-functional equations Proc. Roy. Soc. Edinburgh Sect. A ) [14] T. Matala-aho On the alues of continued fractions: q-series J. Approx. Theory ) [15] T. Matala-aho On irrationality measures of l=0 dl / l j=1 1 + d j r + d 2 j s) J. Number Theory 128 2) 2008) [16] T. Matala-aho V. Merilä On the alues of continued fractions: q-series II Int. J. Number Theory 2 3) 2006) [17] C.F. Osgood On the Diophantine approximation of alues of functions satisfying certain linear q-difference equations J. Number Theory ) [18] I. Shiokawa Rational approximations to the Rogers Ramanujan continued fraction Acta Arith ) [19] Th. Stihl Arithmetische Eigenschaften spezieller Heinescher Reihen Math. Ann ) [20] L.C. Zhang q-difference equations and Ramanujan Selberg continued fractions Acta Arith. 57 4) 1991)