LOWER BOUNDS FOR LINEAR FORMS OF q-series

Transcription

1 LOWER BOUNDS FOR LINEAR FORMS OF q-series Tapani Matala-aho Matematiikan laitos, Oulun Yliopisto, Finland Paris VI, Jussieu, 2 December, 2010

2 q-series Here, we will look q-series which are solutions of q-difference (q-functional) equations P 0 (t)f (q m t) P m (t)f (t) = 0, (1) where P 0 (t),..., P m (t) C p [t] are polynomials, q C p is a parameter and p is a prime or infinity.

3 Generalized q-hypergeometric (basic) series A q-derivation J = J t : JF (t) = F (qt), J = σ q =... (2) An important special case of (1) is {Q(J/q) tp(j)} F (t) = Q(1/q), (3) which is satisfied by F (t) = n=0 n 1 t n P(q k ) Q(q k ). (4) k=0

4 q-factorials q-factorials (q-pochhammer symbols): (a) n = (a; q) n = (1 a)(1 aq) (1 aq n 1 ) (5) (q) n a analogue of n!. (q) n = (q; q) n = (1 q) (1 q n )

5 Basic hypergeometric series q-hypergeometric (basic) series are solutions of ( a1,..., a ) A AΦ B t = b 1,..., b B n=0 (a 1 ) n...(a A ) n (q) n (b 1 ) n...(b B ) n t n. (6) {Q(J/q) tp(j)} F (t) = 0, (7) P(x) = (1 a 1 x) (1 a A x), Q(x) = (1 qx)(1 b 1 x) (1 b B x).

6 q-world/analogues of exponential series q-analogues of exponential series: A q-analogue E q (t) = n=0 of Euler s divergent series: 1 (q) n t n, Ê q (t) = D q (t) = n=0 q (n 2) (q) n t n (8) (q) n t n. (9) n=0 n!t n. (10) n=0

7 q-world q-difference equations E q (qt) = (1 t)e q (t), Ê q (t) = (1 + t)êq(qt). (11) By using (11) one gets the well-known Euler formulae E q (t) = 1 (t) = Ê q (t) = ( t) = 1 k=0 (1 tqk ), (12) (1 + tq k ). (13) k=0

8 q-identities From (12) and (13) we get Ê q (t)e q ( t) = 1 (14) which implies E 1/q,1/q (t)e q,q (qt) = 1 (15)

9 q-world numbers p-adic, p P: n=1 n=0 i=1 n=0 p n 1 p n / Q (16) p n n (1 ± p i ) / Q (17) p n2 n i=1 (1 ± pi ) 2 / Q (18)

10 q-world numbers 1 + p p 2 p / Q (19) (1 + p n ) / Q, (20) n=0 p n n=0 i=1 n (1 + p i ) / Q (21)

11 q-world numbers Real, p Z {0, ±1}: n=1 n=0 i=1 n=0 1 1 p n / Q (22) 1 / Q (23) n 1 ± p i 1 n i=1 (1 ± pi ) 2 / Q (24)

12 q-world numbers p 2 p p / Q (25) (1 + p n ) / Q, (26) n=0 p n n=0 i=1 n (1 + p i ) / Q. (27)

13 q-world numbers n=0 n=0 1 F an+b / Q (28) 1 L an+b / Q (29) where a, b, Z +, (a, b) = 1, F n and L n are the Fibonacci and Lucas numbers, respectively; F 0 = 0, F 1 = 1, L 0 = 2, L 1 = 1.

14 Arithmetic of q-series Amou M., André Y., Bertrand D., Bézivin J-P., Borwein P., Bundschuh P., Di Vizio L., Duverney D., Katsurada M., Merilä V., Nesterenko Yu., Nishioka K., Prévost M., Rivoal T., Stihl Th., Shiokawa I., Waldschmidt M., Wallisser R., Väänänen K., Zudilin W.

15 Notations Let K be an algebraic number field of degree κ over Q. We normalize the absolute value v of K so that if v p, then p v = p 1, if v, then x v = x Q, x Q. By using the re-normalized valuations κv /κ α v = α v, κ v = [K v : Q v ], the product formula has the form α v = 1 α K. v

16 Notations The Height H(α) of α is defined by the formula H(α) = v α v, α v = max{1, α v } 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 }.

17 Notations For any place v of K, q K and q v = 1, we define the number λ = λ q,v = log H(q) log q v. q v < 1 λ q,v 1, (30) d Z {0, ±1}, q = 1/d, λ q, = 1. (31)

18 Stihl/Katsurada Stihl [Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, (1984)] constructed Padé approximations for the above q-hypergeometric series and applied them to the series F (t) = with q < 1. n=0 Katsurada: With usual derivatives. q A(n 2) (q) n (b 1 ) n...(b B ) n t n, 0 B A 2 (32)

19 Stihl Stihl: Suppose q Q, q < 1 satisfies a certain technical condition (L). Let α 1,..., α m Q satisfy α i α j = q n i = j, n Z. (33) Then there exist positive constants C, H 0 such that m k a k i,j F (α i q j ) > C, a A, (34) Hω i=1 j=0 for all k = t (k 0, k i,j ) Z ma+1 {0}, with H = max(h(k), H 0 ) 1, and ω = O(m), a = 1. (35)

20 Condition (L) Usually, the condition (L) restricts the choice of q, when Padé approximations or Thue-Siegel s lemma are used. L = 0. K = Q, v is the infinite place of Q, and q = 1/r, r > 1, r Z. Stihl s case. More generally (later): the cases include e.g. the following examples. L = 1. K = I an imaginary quadratic field, v is the infinite place of I, and q = 1/r, r v > 1, r Z I the ring of integers in I; L = 2. K = Q, v = p, and q = p h, h Z + ; L = 3. for example in K = Q( 5) we may take q 1 = ( 5 1)/2, q = q1, h h Z +.

21 Stihl Stihl s results do not include the q-exponential series E q (t) = n=0 1 (q) n t n, Ê q (t) = But, it does include Tschakaloff function T q (t) = n=0 q (n 2) (q) n t n, (36) q (n 2) t n. (37) n=0

22 BEZIVIN Bezivin J-P. [Independance lineaire des valeurs des solutions trascendantes de certaines equations fonctionnelles, Manuscripta Math. 61 (1988) ] Bezivin [2,3] proved linear independence results for the series F (t) = n=0 t n q D(n 2) n 1, deg Q(x) D. (38) k=0 Q(qk ) and its derivatives over imaginary quadratic field I, when 1/q Z I, q < 1 (case L = 1). Includes both Tschakaloff function T q (t) and q-exponential Ê q (t). But no measure.

23 ANDRÉ André Y. [Séries Gevrey de type arithmétique II, Transcendance sans transcendance, Ann. of Math. (2) 151 (2000) ] Suppose q K, q v < 1, α 1,..., α m K satisfy α i α j = q n i = j, n Z. (39) Put F (t) = T q (t) or F (t) = Ê q (t) and let β 1,..., β m K be such that the v-adic value β 1 F (α 1 ) β m F (α m ) = 0. (40) Then β 1 F (α 1 ) =... = β m F (α m ) = 0. (41)

24 Dimension estimates/tm Matala-aho T. [On Diophantine approximations of the solutions of q-functional equations, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), ] Given a sequence of linear forms R n = P n,1 α P n,m α m, P n,1,..., P n,m K, n N, in m 2 complex or p-adic numbers α 1,..., α m K v with appropriate growth conditions Nesterenko proved a lower bound for the dimension d of the vector space Kα Kα m over K, when K = Q and v is the infinite place.

25 Dimension estimates/tm In the following we shall generalize Nesterenko s dimension estimate over number fields K with appropriate places v, if the lower bound condition for R n is replaced by a determinant condition. Let us have a sequence of linear forms R n = P n,1 α P n,m α m, P n,1,..., P n,m K, n N, (42) in m complex or p-adic numbers α 1,..., α m K v. First we shall study the dimension of the vector space Kα Kα m over K under the following assumptions:

26 Dimension estimates/tm max{ P n,1 w,..., P n,m w } P w (n) w, (43) R n v R v (n) (44) and Δ(n) = det for all n n 0. P n,1... P n,m P n 1,1... P n 1,m..... = 0 (45) P n m+1,1... P n m+1,m

27 Dimension estimates/tm Theorem. Let v be given and let q K satisfy q v < 1. Let P n,1,...,p n,m K be such that the assumptions (43), (44) and (45) are valid with w P w (n) = c a(n) 1 H(q) p(n), c 1, (46) R v (n) = c a(n) 2 q v r(n), c 2 > 0, (47) where c i s are positive constants not depending on n

28 Dimension estimates/tm and such that r(n) (48) a(n)/r(n) 0, r(n)/p(n) h > 1. (49) If at least one of α i s is non-zero, say α 1 = 0, then d = dim K (Kα Kα m ) h λ q. (50) Also a linear independence measure is given for the d linearly independent numbers.

29 Applications As an application we prove that the initial values F (t), F (qt),..., F (q m 1 t) of NF (q m t) = P 1 F (q m 1 t) + P 2 F (q m 2 t) P m F (t), (51) where N = N(q, t), P i = P i (q, t) K[q, t] (i = 1,..., m), generate a vector space of dimension d 2 over K under some conditions for the coefficient polynomials, the solution F (t) and t, q K. Note: The solution F (t) in (51) need not nesessarily to be analytic or continuous.

30 Applications to q-continued fractions Matala-aho T. [On Diophantine Approximations of the Rogers-Ramanujan Continued Fraction, J. Number Theory 45, 1993, ] Matala-aho T. and Merilä V. [On Diophantine approximations of Ramanujan type q-continued fractions, J. Number Theory 129 (2009) ] Irrationality measure results for q-continued fractions generalizing Rogers-Ramanujan continued fraction.

31 Functional equation method/q-iterations Usually, to get good applications we need to accelerate our functional (Padé) approximations. Example: Say, we have a Padé approximation B(t)E q (t) A(t) = t 2n+1 S(t), deg B(t) = deg A(t) = n. (52) But not enough for good lower bound. Operate (52) by J. Then

32 Functional equation method/q-iterations B(qt)E q (qt) A(qt) = q 2n+1 t 2n+1 S(qt), (53) use E q (qt) = (1 t)e q (t) (54) (1 t)b(qt)e q (t) A(qt) = q 2n+1 t 2n+1 S(qt)... (55) (t) K B(q K t)e q (t) A(q K t) = q K(2n+1) t 2n+1 S(q K t). (56) It turns out that the choice K = n is good in this case.

33 Entire/non-entire Let p be a prime or infinite and let q p < 1, then the series F (t) = n=0 including the basic series t n n 1, deg Q(x) = m, Q(1/q) = 0, (57) k=0 Q(qk ) ( ) Φ m 1 (t) = Φ m 1 t = b 1,..., b m 1 n=0 t n (q) n (b 1 ) n...(b m 1 ) n (58) convergences in the unit circle in C p. On the other hand, if q p > 1, then the series (57) and (58) determine entire functions in C p.

34 Entire/non-entire Hence we have two different classes: q p < 1 Entire functions. q p > 1 Non-entire functions. Note, that most of the earlier works deal with the case of entire functions.

35 Dimension estimates/with Amou Amou M. and Matala-aho T. [On Diophantine approximations of the solutions of q-functional equations II, (2004), Submitted, 20 pp] Here we have q v < 1, a non-entire case: We use Siegel s method with the functional equation method. However, because the non-vanishing of the crucial determinant will be proved directly without Shidlovsky s lemma we get the opportunity to make an optimal choise of the parameters coming from Thue-Siegel lemma and the iteration process.

36 Dimension estimates/with Amou Hence, for example, under the assumptions K = Q and 1/q Z {0, ±1} we have dim Q {QF (t) QF (q m 1 t)} > m 1/3 /4 (59) for the functions (57) and (58).

37 Dimension estimates/krattenthaler, Rivoal, Zudilin Krattenthaler C., Rivoal T., Zudilin W. [Series hypergéométriques basiques, fonction q-zeta et séries d Eisenstein, J. Inst. Math. Jussieu 5.1 (2006), ] A q-analogue of zeta-function: ζ q (s) = σ s 1 (n)q n = n=0 m=0 σ s (n) = d s. d n m s 1 q m 1 q m, (60)

38 Dimension estimates/krattenthaler, Rivoal, Zudilin Suppose 1/q Z {0, ±1}, then dim Q {Q + Qζ q (3) + Qζ q (5) Qζ q (2m 1)} > π + o(1) 2 π (2m)1/2 (61)

39 q-analogues of Shidlovskii s and Chudnovsky s lemmas Amou M., Matala-aho T., Väänänen K. [On Siegel-Shidlovskii s theory for q-difference equations. Acta Arith. 127, (2007)] Let K be any field and q K not a root of unity. Take P GL(m, K(z)) and consider a formal power series solution y = t (f 1,..., f m ) K[[z]] m of a linear homogeneous q-difference equation Jy = Py, J = J z. (62)

40 q-analogues of Shidlovskii s and Chudnovsky s lemmas For A = t (A 1,..., A m ) K(z) m we set a linear form m L = A, y := A j f j. (63) j=1 Note that JL = JA, Jy = JA, Py = t PJA, y. (64)

41 q-analogues of Shidlovskii s and Chudnovsky s lemmas Let and define A 1 K(z) m, L 1 = A 1, y (65) A i+1 := t PJA i K(z) m, (66) L i+1 := J(L i ) = A i+1, y i Z +. (67)

42 q-shidlovskii s lemma/i kind approximations Suppose that Jy = Py, y = t (f 1,..., f m ) K[[z]] m (68) dim K(z) (K(z)f 1 (z) K(z)f m ) = m (69) Let L 1 = A 1, y, (70) where A 1 = t (A 11,..., A 1m ) K[z] m {0} and deg A 1j n Z +. If the K(z)-vector space generated by A i (i Z + ) has dimension r, then ord L 1 rn + O(1), (71) where the implied constant in O-symbol depends on the system (43), but not on n.

43 q-shidlovskii s lemma/i kind approximations Corollary: If (71) does not hold for any r < m, then det(a ij ) 1 i,j m = 0. (72) In other words: If we can construct a non-zero linear form with enough big order, then the Corollary holds.

44 q-chudnovsky s lemma/ii kind approximations Shidlovskii s lemma considers rational approximations of first kind while Chudnovsky proved a corresponding result for approximations of second kind. Using our method we have a q-analogue also for Chudnovsky s lemma. Note: Our method may be used for certain Mahler functions, too.

45 On the proof Let F be a free K(z)-algebra generated by u ij (1 i, j m). (Thus we have a rational function field F = K(z, u ij ) with algebraically independent indeterminates u ij over K(z).) Then we extend J to an automorphism of F by JU = PU, U = (u ij ) GL(m, F ), (73) where the constant field C of the difference field (F, J) has an important role.

46 Daniel Bertrand Daniel Bertrand [Multiplicity estimates for q-difference operators, Diophantine Geometry 65-71, CRM Series, 4, Ed. Norm. Pisa, 2007] gave an independent proof for q-shidlovskii s lemma.

47 Applications Let K be an algebraic number field of degree κ over Q, v a place of K and κ v = [K v : Q v ]. Suppose q K, q v < 1 satisfies a certain technical condition (L). Let α 1,..., α m K satisfy Let a(x) = α i α j = q n i = j, n Z. (74) s a ν x ν K[x], a 0 = 0 (75) ν=0 and suppose a(q k ) = 0 for all k Z +.

48 Applications Denote F μν (z) = n=0 where μ = 0, 1,..., s 1; ν = 0, 1,..., l 1. q s(n+1 2 ) n ν a(q) a(q n ) (qμ z) n, (76)

49 Applications Suppose l, m, s Z +. Assume a condition (L) for q. Then the M + 1 = msl + 1 numbers 1, F μν (α j ) K v (77) are linearly independent over K. Further, there exist positive constants C, D, H 0 such that k 0 + k 1 F 11 (α 1 ) k M F s 1,l 1 (α m ) v > C, (78) H ωκ/κv HD(log H) 1/2 for all k = t (k 0,..., k M ) K M+1 {0}, with H = max(h(k), H 0 ) 1, and ω 9sM 2. (79)

50 Applications COROLLARY: Put p P. Now in each of the two sets (1 + kp n ), k = 0, 1,..., p 1, (80) n=1 (1 + kp n ), k = 0, 1,..., p 1, (81) n=1 of p numbers above we have the linear independence of the p numbers over Q with a measure having an exponent ω < 9p 2. (82)

51 Padé type approximations/tm Matala-aho T. [On q-analogues of divergent and exponential series, J. Math. Soc. Japan 61 (2009), ] By modifying Maier-Stihl method explicit Padé type approximations in variable t were constructed for at x = α 1,..., α m. D x (t) = (x; q) n t n (83) n=0 Note the transit to the modified q-exponential series D x (t) = k=0 q (k 2) ( tx) k (t; q) k+1. (84)

52 Applications Put p P. Now in each of the four sets (1 + kp n ), k = 0, 1,..., p 1, (85) n=1 n=1 p n n (1 + kp i ), k = 0, 1,..., p 1, (86) i=1 (1 + kp n ), k = 0, 1,..., p 1, (87) n=1 p n n=1 i=1 n (1 + kp i ), k = 0, 1,..., p 1. (88) of p numbers above we have the linear independence of the p numbers over Q with a measure having an exponent ω < 2p (89)

53 Padé approximations/q-world The q-binomial coefficients are defined by q-binomial theorem: (b, a; q) n = [ n ] = k n k=0 (q; q) n (q; q) k (q; q) n k (90) [ n k ] q (k 2) b n k ( a) k n N, (91) where (b, a; q) 0 = 1 (b, a; q) n = (b a)(b aq)...(b aq n 1 ), n Z +

54 Padé approximations/q-world Stihl [Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, (1984)] Let l, m Z + and α = t (α 1,..., α m ) be given and define σ q,i = σ q,i (l, α) by m ml (α t, w; q 1 ) l = σ q,i w i (92) Then t=1 i=0 ml i=0 for all j {1,..., m}; k {0,..., l 1}. σ q,i (α j q k ) i = 0 (93)

55 Padé approximations/q-world Moreover, by q-binomial theorem, σ q,i = ( 1) i q m(l 2) Σq,ml i (94) holds with Σ q,h = Σ q,h (l, α) = i i m=h [ l i 1 ] [ ] l i m q (i 1 2 )+...+( im 2 ) α i 11 α im m. (95)

56 Padé approximations/q-world F (t) = n=0 d = max{deg P(y), deg Q(y)}. n 1 k=0 P(qk ) n 1 k=0 Q(qk ) tn, (96) Stihl constructed the following explicit type II Padé approximations in variable t for the d series J b F (t), 0 b d 1 at m points.

57 Padé approximations/q-world Let b, l, m, λ N, 0 b < d = max{deg P(y), deg Q(y)} and choose m numbers α 1,..., α m. Put σ q,i = σ q,i (l, α) and ml ml B l,λ (t) = t h b l,λ,h = t ml i [Q; q] i+λ+ l/d 1 σ q,i (97) [P; q] i+λ h=0 i=0 Then B l,λ (t)j b F (α j t) A l,λ,b,j (t) = R l,λ,b,j (t), (98) holds with the parameters [ml, ml + λ 1, ml + l/d + λ]. (99)

58 Padé approximations/q-world This means that the polynomials B l,λ (t) are Padé approximant denominators in variable t for the functions F b,j (t) = J b F (tα j ), b = 0, 1,..., d 1; j = 1,..., m.

59 Modified Maier-Stihl/q-world In the following we will present explicit Padé type approximations in variable t for D x (t) = (x; q) n t n (100) n=0

60 Modified Maier-Stihl/q-world Define q-factorial polynomials n 1 n (x; q) n = (1 xq h ) = s(n, k)x k (101) h=0 k=0 and set s(n, k) = 0, when k < 0 or n < k. Then n s(n, k)( x) n k = q (n 2) x n n N (102) k=0 which plays an essential role in the following modification.

61 Modified Maier-Stihl/q-world Define the following polynomials with ml B l,ν (t) = b l,ν,h (t)t h, (103) h=0 b l,ν,h (t) = ( 1) ml h q (ml+ν 2 ) ( ml+ν h 2 ) (t; q)ml+ν h Σ q,h (104)

62 Modified Maier-Stihl/q-world Then by (101) with b l,ν,h = q m(l 2) B l,ν (t) = ml i+f =H 0 f i+ν ml+ν ml+ν H=0 b l,ν,h t H (105) q (ml+ν 2 ) ( i+ν 2 ) s(i + ν, f )σq,i (106)

63 Modified Padé approximations Now we get the following modified Padé type approximations of the second kind for the m functions D αj (t), (j = 1,..., m) in variable t. Let l, ν N and j = 1,..., m, then B l,ν (t)d αj (t) + A l,ν,j (t) = L l,ν,j (t) (107) with [ml + ν, ml + ν 1, (m + 1)l + ν] (108) By (108) we have a diagonal type Padé approximation with the free parameter ν.

64 Modified Maier-Stihl/q-world Further where and A l,ν,j (t) = a l,ν,j,n = ml+ν 1 N=0 H+n=N a l,ν,j,n t N, (109) b l,ν,h ( α j ; q) n, (110)

65 Modified Maier-Stihl/q-world L l,ν,j (t) = t (m+1)l+ν q (ml+ν 2 )+m( l 2)+νl ( α j ; q) l α ν j S l,ν,j (t). (111) where with S l,ν,j (t) = s l,ν,j,k t k (112) k=0 s l,ν,j,k = q kν ( α j q l ; q) k m t=1 (α t, α j q k+1 ; q) l. (113)

66 Modified Maier-Stihl/Proof The expansion holds with B l,ν (t)d αj (t) = r N t N, (114) N=0 r N = b l,ν,h ( α j ; q) n. (115) H+n=N

67 Modified Maier-Stihl/Proof Set N = ml + ν + a 0 a l 1 then r N = q m(l 2) ml i+ν q (ml+ν 2 ) ( i+ν 2 ) s(i + ν, f )σq,i ( α j ; q) i+ν f +a = i=0 f =0 q m(l 2) ml σ q,i ( α j ; q) a q (ml+ν 2 ) ( i+ν 2 ) i=0 i+ν s(i + ν, f )( α j q a ; q) i+ν f (116) f =0

68 Modified Maier-Stihl/Proof Here the inner f -sum is evaluated by (102) and so ml r ml+ν+a = q m(l 2)+( ml+ν 2 ) ( αj ; q) a σ q,i (α j q a ) i+ν = q m(l 2)+( ml+ν 2 )+aν ( α j ; q) a α ν j for any 0 a l 1. i=0 m (α t, α j q a ; q 1 ) l = 0 (117) t=1

69 Modified Maier-Stihl/Proof Next we consider the case a = l + k, k N. Then r N = r (m+1)l+ν+k = q m(l 2)+( ml+ν 2 )+(l+k)ν ( α j ; q) l+k α ν j m (α t, α j q k+1 ; q) l. (118) t=1