ARITHMETIC WITH HYPERGEOMETRIC SERIES

Transcription

1 ARITHMETIC WITH HYPERGEOMETRIC SERIES Tapani Matala-aho Matematiikan laitos, Oulun Yliopisto, Finland Stockholm 2010 May 26, Finnish-Swedish Number Theory Conference May, 2010

2 Arithmetic Motivation An interesting part of Number Theory is involved with a question of arithmetic nature of explicitly defined numbers. -Irrationality -Linear independence over a field -Transcendence

3 Arithmetic Motivation Even more interesting and challenging with a quantitative setting. -Irrationality measure -Linear independence measure -Transcendence measure

4 Generalized Hypergeometric series Let P(y) and Q(y) = 0(y) be polynomials and define generalized hypergeometric series F (t) = and q-hypergeometric series F q (t) = n 1 k=0 P(k) n 1 k=0 Q(k)tn (1) n 1 k=0 P(qk ) n 1 k=0 Q(qk ) tn (2)

5 Classical hypergeometric series Pochhammer symbol (generalized factorial) (a) 0 = 1, (a) n = a(a + 1) (a + n 1) (3) Hypergeometric series ( a1,..., a ) A AF B t = b 1,..., b B (1) n = n! n Z +. (4) (a 1 ) n (a A ) n n!(b 1 ) n (b B ) n t n (5)

6 Gauss hypergeometric series Gauss hypergeometric series 2F 1 ( a, b c ) t = (a) n (b) n n!(c) n t n. (6)

7 Gauss hypergeometric series/cases Geometric series Logarithm series Binomial series: Arcustangent: 2F 1 ( 1, 1 1 2F 1 ( 1, 1 2 2F 1 ( 1, α 2 2F 1 ( 1, 1/2 3/2 ) ( 1 t = 1 F 0 ) t = ) log(1 t) t = = t ) t = (1 t) α = ) t 2 = arctan t t = t n (7) 1 n + 1 tn (8) ( ) α ( t) n (9) n ( 1) n 2n + 1 t2n+1 (10)

8 Gauss hypergeometric series/cases Jacobi polynomials: 2F 1 ( n, α + β + n + 1 α + 1 Legendre polynomials: 2F 1 ( n, n ) t = Tsebycheff and Gegenbauer polynomials. n! (α + 1) n P (α,β) n (1 2t) (11) ) t = P n (1 2t) (12)

9 Gauss hypergeometric series/cases Elliptic integrals: K(t) = π/2 0 dθ 1 1 t 2 sin 2 θ = dx (1 x 2 )(1 t 2 x 2 ) 0 (13) E(t) = π/ t 1 t 2 sin 2 θdθ = 2 x 2 dx (14) 0 1 x 2 2F 1 ( 1/2, 1/2 1 2F 1 ( 1/2, 1/2 1 ) t 2 = 2 K(t) (15) π ) t 2 = 2 E(t) (16) π

10 Other Exponent: Bessel function J a : 0F 0 ( 0F 1 ( α ) t = exp(t) = 1 n! tn (17) ) t = Γ(α)(it) α 1 J α 1 (2it 1/2 ) (18) Euler s series 2F 0 ( 1, 1 ) t = n!t n, (19)

11 Classical numbers/irrationality e = 1 n! / Q (20) log 2 = π = 4 ( 1) n n + 1 ( 1) n 2n + 1 / Q (21) / Q (22)

12 Classical numbers/linear independence m {0, 1, 2,...}. Hermite: dim Q {Qe Qe m } = m + 1 (23)

13 Classical numbers/linear independence Apéry, Rivoal, Ball, Zudilin: dim Q {Q + Qζ(3) + Qζ(5) Qζ(2m + 1)} = 2, m = 1; (24) 2 log(2m + 1) log 2 (25) dim Q {Q + Qζ(5) + Qζ(7) + Qζ(9) + Qζ(11)} 2 (26)

14 Classical numbers/linear independence Conjecture: dim Q {Q + Qπ + Qζ(3) + Qζ(5) Qζ(2m + 1)} = m + 2 (27) and more generally it is conjectured: The numbers π, ζ(3), ζ(5),..., ζ(2m + 1) (28) are algebraically independent.

15 Classical numbers/p-adic meaning Euler s divergent series (Wallis series) 2F 0 ( 1, 1 ) ±1 = Conjecture: Transcendental. Note 2F 0 ( 1, 1 n!(±1) n Q?? (29) ) 1 = n n! Q (30)

16 Basic hypergeometric series q-series factorials (q-pochhammer symbols): (a) n = (a; q) n = (1 a)(1 aq) (1 aq n 1 ) (31) q-hypergeometric (basic) series ( a1,..., a ) A AΦ B t = b 1,..., b B (q) n = (q; q) n = (1 q)...(1 q n ) (a 1 ; q) n...(a A ; q) n (q; q) n (b 1 ; q) n...(b B ; q) n t n. (32)

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

18 q-world numbers p-adic, p P: n=1 n=1 i=1 p n 1 p n / Q (33) p n / Q (34) n 1 ± p i p n2 n j=1 (1 ± pj ) 2 / Q (35)

19 q-world numbers 1 + p p 2 p / Q (36) (1 + kp n ), k = 1,..., p 1, (37) n=1 p n n=1 i=1 n (1 + kp i ), k = 1,..., p 1, (38) For the set (37) [Väänänen] gave dim Q = p (39) True also for the set (38).

20 q-world numbers Real, p Z {0, ±1}: n=1 n=1 i=1 1 1 p n / Q (40) 1 / Q (41) n 1 ± p i 1 n j=1 (1 ± pj ) 2 / Q (42)

21 q-world numbers p 2 p p / Q [Bundschuh] (43) (1 + kp n ), k = 0, 1,..., p 1, (44) n=1 p n n=1 i=1 n (1 + kp i ), k = 0, 1,..., p 1. (45) For the set (37) [Väänänen] gave dim Q = p (46) True also for the set (38).

22 q-world numbers 1 F an+b / Q (47) 1 L an+b / Q (48) where a, b, Z +, F n and L n are the Fibonacci and Lucas numbers, respectively; F 0 = 0, F 1 = 1, L 0 = 2, L 1 = 1. [André-Jeannin]: a = 1; [Bundschuh+Väänänen] with a measure. [Prévost+T.M.]: a, b 1 with irrationality measures; [Merilä].

23 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 (49) for every M/N Q with H = max{ M, N } H 0 (ε).

24 Irrationality measures of explicit numbers μ(e) = 2 Classical (50) μ(log 2) [Rukhadze] (51) μ(log 3) [Salikhov] (52) μ(π) [Hata] (53) μ(ζ(2)) [Rhin+Viola] (54) μ(ζ(3)) [Rhin+Viola] (55)

25 Irrationality measures of explicit numbers τ 1 = F 1 + F 2 + F , μ(τ 1) = 2 (56) τ 2 = L 1 + L 2 + L , μ(τ 2) = 2 (57) τ i M N C N 2+D/ log N (58)

26 Linear forms Let Θ C p be a number to be studied. a) p =. I an imaginary quadratic field and Z I ring of integers. b) p P = {2, 3, 5,...}. I = Q.

27 Linear forms In the following theorems put Q(n) = e a(n), R(n) = e b(n) (59) where or and a(n) = an, b(n) = bn (classical) (60) a(n) = an log n, b(n) = bn log n (classical) (61) a(n) = an 2, b(n) = bn 2 (q-world). (62)

28 Linear forms Assume that R n = B n Θ A n n N (63) are numerical approximation forms satisfying B n, A n Z I (64) B n A n+1 A n B n+1 = 0, (65) B n Q(n), and also (66) A n Q(n), if p = (67) R n p R(n) (68) for all n n 0 with some positive a and b and a < b, if p =.

29 Linear forms/axiomatic Let the above assumptions be valid. Then for every ε > 0 there exists a constant H 0 = H 0 (ε) 1 such that Θ M N > H μ ε (69) p for all M, N Z I with H H 0, where (by folkflore) μ = μ = 1 + a/b, H = N, if p =, (70) b, H = max{ M, N }, if p P. (71) b a Kalle Leppälä (Master thesis): Axiomatic for more general a(n) and b(n).

30 Linear forms/over algebraic numbers/several variables -Use valuations of a number field with product formula. -Several variables with larger determinants. Need a construction of appropriate Linear Forms.

31 Padé approximations/classical case First we will study the classical series F (t) with it s derivativies Δ b F (t), where Δ = t d dt. Denote d = max{deg P(y), deg Q(y)} and let d, m Z + and the numbers α 1,..., α m be given. We start by giving explicit type II Padé approximations for the series Δ b F (tα j ), b = 0, 1,..., d 1; j = 1,..., m. (72) Our construction is based on a product expansion a la Maier [Potenzreihen irrationalen Grenzwertes. J. Reine Angew. Math. 156, (1927)]

32 Maier s product formula Let l, m Z + and α = t (α 1,..., α m ) be given and define σ i = σ i (l, α) by Then m ml (α t w) l = σ i w i. (73) t=1 i=0 ml i=0 for all t {1,..., m}; k {0,..., l 1}. σ i i k α i t = 0 (74)

33 Maier s product formula Moreover σ i = ( 1) i i i m=i ( l i 1 ) ( ) l α l i 1 1 αm l im. (75) i m

34 Padé approximations/classical case Let b, d, l, m, λ N, b < d and choose m numbers α 1,..., α m. Put ml B l,λ (t) = t ml i σ i (l, α) [Q] i+λ+ l/d 1. (76) [P] i+λ i=0 Then B l,λ (t)δ b F (α j t) A l,λ,b,j (t) = R l,λ,b,j (t), (77) where deg t B l,λ (t) = ml, deg t A l,λ,b,j (t) ml + λ 1 (78) ord R l,λ,b,j(t) ml + l/d + λ. (79) t=0

35 Padé approximations/classical case Thus we have a gap of lenght l/d in the power series expansion B l,λ (t)δ b F (α j t) = A l,λ,b,j (t) + R l,λ,b,j (t). (80) The polynomials B l,λ (t) are Padé approximant denominators in variable t for the functions F b,j (t) = Δ b F (tα j ), b = 0, 1,..., d 1; j = 1,..., m. Also we say that (77 79) define a Padé approximation with the degree and order parameters [deg t B, deg t A, ord t=0 R ] = [ml, ml + λ 1, ml + l/d + λ] (81)