RAMANUJAN TYPE q-continued FRACTIONS

Transcription

1 RAMANUJAN TYPE q-continued FRACTIONS Tapani Matala-aho 19th Czech and Slovak Number Theory Conference August 31 September 4, 2009 Hradec nad Moravicí, Czech Republic

2 References [MA1] Matala-aho T., On Diophantine Approximations of the Rogers-Ramanujan Continued Fraction, J. Number Theory 45, 1993, [MA2] Matala-aho T., On the values of continued fractions: q-series, J. Approx. Th. 124 (2003) [MAME1] Matala-aho T. and Merilä V., On the values of continued fractions: q-series II, Int. J. Number Theory 2 (2006) [MAME2] Matala-aho T. and Merilä V., On Diophantine approximations of Ramanujan type q-continued fractions, J. Number Theory 129 (2009)

3 CONTINUED FRACTIONS b 0 + a n K = b 0 + a 1 n=1 b n b 1 + a 2 a 3 b 2 + b = (1) Simple continued fraction b 0 + a 1 b 1 + a 2 b (2) [b 0 ; b 1, b 2,...] = b 0 + K n=1 1 b n (3)

4 NOTATIONS For any p of the set { } P, P = {p Z + p is a prime} the notation = will be used for the usual absolute value of C = C and p for the p-adic valuation of the p-adic field C p, the completion of the algebraic closure of Q p, normalized by p p = p 1.

5 VALUES By the value of the continued fraction in C p we mean the limit b 0 + K n=1 a n b n (4) A n lim n B n of the convergents 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 values A 0 = b 0, A 1 = b 0 b 1 + a 1, B 0 = 1, B 1 = b 1.

6 IRRATIONALITY MEASURE By an effective irrationality measure (exponent) of a given number α C p we mean a number μ = μ(α) 2 which satisfies the condition: for every ε > 0 there exists an effectively computable constant H 0 (ε) 1 such that α M N > 1 p H μ+ε (5) for every M/N Q with H = max{ M, N } H 0 (ε).

7 ABSTRACT We shall consider arithmetical properties of the q-continued fractions q sn t s (S 0 + S 1 q n t S h q hn t h ) K n=1 T 0 + T 1 q n t T l q ln t l deg=s+h deg=l where S i, T i, q K, q v < 1, and some related continued fractions where v is a fixed valuation of an algebraic number field K and s, h, l N 0. In particular, we get sharp irrationality measures for certain Ramanujan, Ramanujan-Selberg, Eisenstein and Tasoev continued fractions.

8 ROGERS-RAMANUJAN 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 RR(q, t) = 1 + qt q 2 t q 3 t deg=1+0=1 deg=0 (6) in archimedean imaginary quadratic fields, see Bundschuh [8], Osgood [20], Shiokawa [22] and Stihl [23]. Matala-aho [15] considered approximations also in other valuations and proved some higher degree quantitative irrationality results.

9 ROGERS-RAMANUJAN [MA1] RR(1/5, t) = 1 + t t t t t / Q( m), (7) for any m Q and t Q. Moreover RR(1/5, t), has effective irrationality measure μ(rr(1/5, t)) = 2 (8)

10 ROGERS-RAMANUJAN RR with an effective irrationality measure ( ( ) 5 1)/2, t / Q( 5), (9) μ(rr( 5 1)/2, t)) = 2 2 (10)

11 ROGERS-RAMANUJAN Further, if p 17 is a prime number and t Q, then the p-adic number for any m Z. Moreover RR(p, t) = 1 + tp tp 2 tp / Q( m), (11) μ(rr(p, t)) = 2 (12)

12 THUE-SIEGEL S METHOD [Amou M. and Matala-aho T. (2001)] (tq n ) s (S 0 + S 1 q n t S h q hn t h ) K n=1 T 0 + T 1 q n t T l q ln t l (13) with S i, T i K, q p < 1, s 1. Example the Watson-Ramanujan continued fraction WR(q) = K n=1 q n + q 2n 1 has an irrationality measure deg=1+1=2 deg=0 (14) μ(wr(q)) 61, q = 1/d, d Z {0, ±1}.

13 RAMANUJAN [MAME2] For example, for the both q-fractions K n=1 q 2n 1 + q n, K n=1 q 2n 1 deg=2+0=2 1 + q n deg=1 (15) we get the irrationality measure μ = 2 (16) if IP happens I) q = 1/d, d Z {0, ±1}, p = (17) or P) q = p P (18)

14 IP happens If IP happens with Then we say μ = a (19) μ IP = a (20)

15 q-factorials q-factorials (a) 0 = (a; q) 0 = 1 (a) n = (a; q) n = (1 a)(1 aq)... (1 aq n 1 ) n Z + (a; q) = (1 aq n ) n=0

16 RAMANUJAN Note that, concerning the latter continued fraction in (15) a deep claim 1 q q 1 + q q 3... q 3 q 5 = (q2 ; q 3 ) (q; q 3 ) (21) made by Ramanujan is proved in [2] and again quite recently in [4] and [7].

17 RAMANUJAN-SELBERG S 1 (q) = 1 q q + q 2 q 3 q 2 + q with = ( q2 ; q 2 ) ( q; q 2 ) (22) μ IP (S 1 (q)) = 3 (23) Take first even contraction.

18 RAMANUJAN-SELBERG S 2 (q) = 1 q + q 2 q 4 q 3 + q 6 q 8 q 5 + q 10 q with = (q; q8 ) (q 7 ; q 8 ) (q 3 ; q 8 ) (q 5 ; q 8 ), μ IP (S 2 (q)) = 3 (24)

19 RAMANUJAN-SELBERG S 3 (q) = 1 q + q 2 q 2 + q 4 q 3 + q deg=1+1=2 deg=0 with = (q; q2 ) (q 3 ; q 6 ) 3, (25) μ IP (S 3 (q)) 61 (26)

20 EISENSTEIN E 1 (q) = 1 q q 3 q q 5 q 7 q 3 q 9 q 11 q with = q n2, (27) n=0 μ IP (E 1 (q)) = 3 (28) A better is available.

21 EISENSTEIN E 2 (q) = 1 q q + 1 q q q = ( 1) n q n2, (29) n=0 q 3 q 5 q 7 A much better is available. μ IP (E 2 (q)) 61 (30)

22 TASOEV Next, we consider the following Tasoev s continued fractions T 1 (u, v, a) = ua + va 2 + ua 3 + va 4 + ua , (31) T 2 (u, v, a, b) = ua + vb + ua 2 + vb 2 + ua (32) Komatsu has studied evaluations of numerous variants of T 1 and T 2 e.g. in [11], [12], [13], [14]. Later we will show how we may evaluate T 1 and T 2 simply by using the well-known identities from the theory of q-continued fractions, such as (46) of the Rogers-Ramanujan continued fraction.

23 TASOEV μ IP (T 1 (u, v, a)) = μ IP (T 2 (u, v, a, b)) = 2 (33) where u, v Q and a, b Z {0, ±1}.

24 FIBOLOGY Finally, we shall study certain continued fractions W = W 1 + W 2 + W (34) where the partial denominators satisfy a second order recurrence. Let (F n ), F 0 = 0, F 1 = 1 denote the Fibonacci sequence. If we set F = F 1 + F 3 + F , (35) then μ(f ) 61.

25 MEASURE BY DEGREES The above are applications of the following: Suppose s 1. A) Let s > h, s > l. If s + h 2l, then μ IP (α) = 2s s h If s + h < 2l, then μ IP (α) = s s l B) Let s = h or s = l. Then (36) (37) μ IP (α) 61 (38)

26 MORE EXAMPLES a) Let b, d Q, d > 0, e, q I and p =. If q p < 1 and d 2 + 4b = (is a square of a rational number), then μ I ( K n=1 b d + eq n ) = 2 (39)

27 MORE EXAMPLES b) Let b, d Q, e, q K = Q( d 2 + 4b) and p =. If q v < 1, d > 0, d 2 + 4b > 0 and d 2 + 4b =, then μ I ( K n=1 b d + eq n ) = 2 2 (40) NOTE. In the cases a) and b) we have s = 0!!

28 DEVELOPE Suppose F (t) satisfies a functional equation F (t) = T (t)f (qt) + V (qt)f (q 2 t) (41) Then F (t) F (qt) = T (t) + V (qt) F (qt)/f (q 2 t) = (42) V (qt) T (t) + T (qt) + V (q 2 t) F (q 2 t)/f (q 3 t) =... (43)

29 DEVELOPE Denote Question: G(q, t) = T (t) + K n=1 V (q n t) T (q n t) = T (t) + V (qt) T (qt) + V (q2 t) T (q 2 t)+... (44) F (t) = G(q, t)? (45) F (qt) It depends!

30 DEVELOPE As an example, we give the well-known evaluation F (t) F (qt) = 1 + qt q 2 t q 3 t (46) of the Rogers-Ramanujan continued fraction where F (t) = n=0 q n2 (q) n t n, q p < 1, (47) which satisfies F (t) = F (qt) + qtf (q 2 t) (48)

31 MAIN THEOREM Let S(t) = S 0 + S 1 t S h t h, T (t) = T 0 + T 1 t T l t l C p [t] Fix G(q, t) = T (t) + K n=1 (tq n ) s S(q n t) T (q n t) deg=s+h deg=l (49) { l A = max 2, s + h }, B = s 4 2. (50) Generally for good results we need B > A. Thus we need s > l, s > h.

32 VALUATIONS, HEIGHTS Let K be an algebraic number field of degree κ over Q, v its place and K v the corresponding completion. If the finite place v of K lies over the prime p, we write v p, for an infinite place v of K we write v. Further, the notation I is used for an imaginary quadratic field. By using the normalized valuations κv /κ α v = α v, κ v = [K v : Q v ], the Height H(α) of α K is defined by the formula H(α) = v α v, α v = max{1, α v }

33 VALUATIONS, HEIGHTS and the Height H(α) of vector α = (α 1,..., α m ) K m is given by H(α) = v α v, α v = max i=1,...,m {1, α i v }. A characteristic λ = λ q = will also be used in the sequel. log H(q) log q v, q v = 1, q K

34 MAIN THEOREM Case 1): When B + λa > 0, we set μ = μ 1 = B B + λa (51) Case 2): Whereas, when B + λa = 0 and λ = 1 and s 1, we have μ 2 (64s s + 21)/3. (52) In particular, by choosing s = 1 we obtain μ 2 61.

35 MAIN THEOREM Let v be a place of K and let q, t K satisfy B + λa 0, q v < 1 and let S(t), T (t) K[t] satisfy S 0 T 0 = 0, S 0 v 1, s 1 and S(q k t)t (q k t) = 0 for all k N. Then, there exist positive constants C i, D i and H i, i = 1, 2, such that G(q, t) β v > for any β K with H = max{h(β), H i }. C i H κμ i /κ v +D i (log H) 1/2 (53)

36 EXAMPLES Let h = 0 and s 2l > 0, then B/A = 2 gives an irrationality measure μ 1 = 2/(2 + λ) for G(q, t) for all q K satisfying 2 < λ 1. Now, λ = 1 if q = 1/d, d Z {0, ±1}, v = (54) and q = p, p P, v = p (55) Thus μ = 2

37 EXAMPLES and especially G(1/d, t) as well as the p-adic number G(p l, t) (p is a prime, l Z + ) have an irrationality measure μ 1 = 2. When K = I, v is the infinite place of K and q = 1/d, d v > 1, where d Z K, we obtain μ 1 = 2, κ = κ v = 2.

38 EXAMPLES a) Let b, d Q, d > 0, e, q I and v. If q v < 1 and d 2 + 4b is a square of a rational number, then μ 1 ( K n=1 b d + eq n ) = λ. (56) NOTE. In the cases a) and b) we have s = 0!!

39 EXAMPLES b) Let b, d Q, e, q K = Q( d 2 + 4b) and v. If q v < 1, d > 0, d 2 + 4b > 0 and d 2 + 4b is not a square of a rational number, then κ = 2, κ v = 1 and μ 1 ( K n=1 b d + eq n ) = λ. (57)

40 EXAMPLES The continued fraction K n= eq n (58) with q = ( 5 1)/2 belongs to the case b) where λ = 1, μ 1 = 2, and thus K n= eq n β > v C H 4+D(log H) 1/2 (59) for any β K with H = max{h(β), H 1 }. In particular, the value of continued fraction (58) is not in Q( 5).

41 EXAMPLES In [19], it is proved that d + K n=1 b + cq n d + eq n = α + K n=1 cq n + eβq 2n 1 α + (e β)q n (60) where b, c, d, e, q K, q < 1 and α = d + d 2 + 4b 2, β = d d 2 + 4b, β < α. 2

42 EXAMPLES Here, we note that the formula (60) generalizes the transformation formula 1 + K n=1 k + q n 1 = α + K n=1 q n α βq n (61) where q C, q < 1, k R, k > 1/4, α = ( k ) /2 and β = ( k ) /2, originally considered by Ramanujan (for more details, see [5]).

43 REFERENCES Amou M. and Matala-aho T., Arithmetical properties of solutions of linear second order q-difference equations, Number theoretic methods (Iizuka, 2001) Dev. Math. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp Andrews G. E., Berndt B. C., Jacobsen L., Lamphere R. L., The continued fractions found in the unorganized portions of Ramanujan s Notebooks, Mem. Amer. Math. Soc. 99 (1992), no. 477, vi+71 pp. Andrews G. E. and Bowman D., A full extension of the Rogers-Ramanujan continued fraction, Proc. Amer. Math. Soc. 123, 1995, pp

44 REFERENCES Andrews, G. E., Berndt B. C., Sohn J., Yee A. J., Zaharescu A., On Ramanujan s continued fraction for (q 2 ; q 3 ) /(q; q 3 ), Trans. Amer. Math. Soc. 355 (2003), no. 6, Andrews, G. E., Berndt B. C., Ramanujan s lost notebook. Part I., Springer, New York, 2005, xiv+437 pp. Berndt B. C., Ramanujan s Notebooks, Part V, Springer-Verlag, New York 1998

45 REFERENCES Bowman D., Mclauglin J. Wyshinski N. J., A q-continued fraction, Int. J. Number Theory 2 (2006), no. 4, Bundschuh P., Ein Satz uber ganze Funktionen und Irrationalitatsaussagen, Invent. Math. 9, 1970, Duverney D, Nishioka K, Nishioka K and I. Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 73, 1997,

46 REFERENCES Folsom A., Modular forms and Eisenstein s continued fractions, J. Number Theory 117 no. 2, 2006, Komatsu T.,On Hurwitzian and Tasoev s continued fractions, Acta Arith. 107 (2003), no. 2, Komatsu T.,On Tasoev s continued fractions, Math. Proc. camb. Soc. 134, 2003, 1-12.

47 REFERENCES Komatsu T.,Tasoev s continued fractions and Rogers-Ramanujan continued fractions, J. Number Theory, 109, 2004, Komatsu T.,Hurwitz and Tasoev s continued fractions, Monatsh. Math. 145, 2005, Matala-Aho T., On Diophantine Approximations of the Rogers-Ramanujan Continued Fraction, J. Number Theory 45, 1993,

48 REFERENCES Matala-aho T., On Diophantine approximations of the solutions of q-functional equations, Proc. Roy. Soc. Edinburgh Sect. A 132, 2002, Matala-aho T., On the values of continued fractions: q-series, J. Approx. Th. 124, 2003, Matala-aho T., On irrationality measures of l=0 d l / l j=1 (1 + d j r + d 2j s), to appear in J. Number Theory.

49 REFERENCES Matala-aho T., Merilä V., On the values of continued fractions: q-series II, Int. J. Number Theory 2 no. 3, 2006, Osgood C. F., On the Diophantine Approximation of Values of Functions Satisfying Certain Linear q-difference Equations, J. Number Theory 3, 1971, Perron O., Über einen Satz des Herrn Poincaré, J. Reine Angew. Math. 136, 1909, 17 37

50 REFERENCES Shiokawa I., Rational approximations to the Rogers-Ramanujan continued fraction, Acta Arith. 50, 1988, Stihl Th., Arithmetische Eigenschaften spezieller Heinescher Reihen, Math. Ann. 268, 1984, Zhang, L. C., q-difference equations and Ramanujan-Selberg continued fractions, Acta Arith. 57 no. 4, 1991,