arithmetic with hypergeometric seriescc.oulu.fi/~tma/stockholmslides.pdf · arithmetic with...
TRANSCRIPT
ARITHMETIC WITH HYPERGEOMETRIC
SERIES
Tapani Matala-aho
Matematiikan laitos, Oulun Yliopisto, Finland
Stockholm 2010 May 26, Finnish-Swedish Number Theory
Conference 26-28 May, 2010
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
Arithmetic Motivation
Even more interesting and challenging with a quantitative setting.
-Irrationality measure
-Linear independence measure
-Transcendence measure
Generalized Hypergeometric series
Let P(y) and Q(y) ∕= 0(y) be polynomials and define generalized
hypergeometric series
F (t) =∞∑n=0
∏n−1k=0 P(k)∏n−1k=0Q(k)
tn (1)
and q-hypergeometric series
Fq(t) =∞∑n=0
∏n−1k=0 P(qk)∏n−1k=0Q(qk)
tn (2)
Classical hypergeometric series
Pochhammer symbol (generalized factorial)
(a)0 = 1, (a)n = a(a + 1) ⋅ ⋅ ⋅ (a + n − 1) (3)
(1)n = n! n ∈ ℤ+. (4)
Hypergeometric series
AFB
(a1, ..., aAb1, ..., bB
∣∣∣ t) =∞∑n=0
(a1)n ⋅ ⋅ ⋅ (aA)nn!(b1)n ⋅ ⋅ ⋅ (bB)n
tn (5)
Gauss’ hypergeometric series
Gauss’ hypergeometric series
2F1
(a, b
c
∣∣∣ t) =∞∑n=0
(a)n(b)nn!(c)n
tn. (6)
Gauss’ hypergeometric series/cases
Geometric series
2F1
(1, 1
1
∣∣∣ t) = 1F0
(1
∗
∣∣∣ t) =∞∑n=0
tn (7)
Logarithm series
2F1
(1, 1
2
∣∣∣ t) = − log(1− t)
t=∞∑n=0
1
n + 1tn (8)
Binomial series:
2F1
(1,−�
2
∣∣∣ t) = (1− t)� =∞∑n=0
(�
n
)(−t)n (9)
Arcustangent:
2F1
(1, 1/2
3/2
∣∣∣ −t2) =arctan t
t=∞∑n=0
(−1)n
2n + 1t2n+1 (10)
Gauss’ hypergeometric series/cases
Jacobi polynomials:
2F1
(−n, � + � + n + 1
� + 1
∣∣∣ t) =n!
(� + 1)nP(�,�)n (1− 2t) (11)
Legendre polynomials:
2F1
(−n, n + 1
1
∣∣∣ t) = Pn(1− 2t) (12)
→ Tsebycheff and Gegenbauer polynomials.
Gauss’ hypergeometric series/cases
Elliptic integrals:
K (t) =
∫ �/2
0
d�√1− t2 sin2 �
=
∫ 1
0
dx√(1− x2)(1− t2x2)
(13)
E (t) =
∫ �/2
0
√1− t2 sin2 �d� =
∫ 1
0
√1− t2x2√1− x2
dx (14)
2F1
(1/2, 1/2
1
∣∣∣ t2) =2
�K (t) (15)
2F1
(1/2,−1/2
1
∣∣∣ t2) =2
�E (t) (16)
Other
Exponent:
0F0(∗∗
∣∣∣ t) = exp(t) =∞∑n=0
1
n!tn (17)
Bessel function Ja:
0F1( ∗�
∣∣∣ t) = Γ(�)(it)�−1J�−1(2it1/2) (18)
Euler’s series
2F0
(1, 1
∗
∣∣∣ t) =∞∑n=0
n!tn, (19)
Classical numbers/irrationality
e =∞∑n=0
1
n!/∈ ℚ (20)
log 2 =∞∑n=0
(−1)n
n + 1/∈ ℚ (21)
� = 4∞∑n=0
(−1)n
2n + 1/∈ ℚ (22)
Classical numbers/linear independence
m ∈ {0, 1, 2, ...}.
Hermite:
dimℚ{ℚe0 + ...+ ℚem} = m + 1 (23)
Classical numbers/linear independence
Apery, Rivoal, Ball, Zudilin:
dimℚ{ℚ + ℚ�(3) + ℚ�(5) + ...+ ℚ�(2m + 1)}
= 2, m = 1; (24)
≥ 2
3
log(2m + 1)
1 + log 2(25)
dimℚ{ℚ + ℚ�(5) + ℚ�(7) + ℚ�(9) + ℚ�(11)} ≥ 2 (26)
Classical numbers/linear independence
Conjecture:
dimℚ{ℚ + ℚ� + ℚ�(3) + ℚ�(5) + ...+ ℚ�(2m + 1)}
= m + 2 (27)
and more generally it is conjectured: The numbers
�, �(3), �(5), ..., �(2m + 1) (28)
are algebraically independent.
Classical numbers/p-adic meaning
Euler’s divergent series (Wallis series)
2F0
(1, 1
∗
∣∣∣ ±1
)=∞∑n=0
n!(±1)n ∈ ℚ ?? (29)
Conjecture: Transcendental.
Note
2F′0
(1, 1
∗
∣∣∣ 1
)=∞∑n=0
n ⋅ n! ∈ ℚ (30)
Basic hypergeometric series
q-series factorials (q-Pochhammer symbols):
(a)n = (a; q)n = (1− a)(1− aq) ⋅ ⋅ ⋅ (1− aqn−1) (31)
(q)n = (q; q)n = (1− q)...(1− qn)
q-hypergeometric (basic) series
AΦB
(a1, ..., aAb1, ..., bB
∣∣∣ t) =∞∑n=0
(a1; q)n...(aA; q)n(q; q)n(b1; q)n...(bB ; q)n
tn. (32)
Arithmetic of q-series
Amou M., Andre Y., Bertrand D., Bezivin, Borwein P., Bundschuh
P., Duverney D., Katsurada M., Merila V., Nesterenko Yu.,
Nishioka K., Prevost M., Rivoal T., Stihl Th., Shiokawa I.,
Waldscmidt M., Wallisser R., Vaananen K., Zudilin W.
q-world numbers
p-adic, p ∈ ℙ:∞∑n=1
pn
1− pn/∈ ℚ (33)
∞∑n=1
pn
n∏i=1
1± pi/∈ ℚ (34)
∞∑n=0
pn2∏n
j=1(1± pj)2/∈ ℚ (35)
q-world numbers
1 +p
1 +
p2
1 +
p3
1 + . . ./∈ ℚ (36)
∞∏n=1
(1 + kpn), k = 1, ..., p − 1, (37)
∞∑n=1
pnn∏
i=1
(1 + kpi ), k = 1, ..., p − 1, (38)
For the set (37) [Vaananen] gave
dimℚ = p (39)
True also for the set (38).
q-world numbers
Real, p ∈ ℤ ∖ {0,±1}:
∞∑n=1
1
1− pn/∈ ℚ (40)
∞∑n=1
1n∏
i=11± pi
/∈ ℚ (41)
∞∑n=0
1∏nj=1(1± pj)2
/∈ ℚ (42)
q-world numbers
1 +p−1
1 +
p−2
1 +
p−3
1 + . . ./∈ ℚ [Bundschuh] (43)
∞∏n=1
(1 + kp−n), k = 0, 1, ..., p − 1, (44)
∞∑n=1
p−nn∏
i=1
(1 + kp−i ), k = 0, 1, ..., p − 1. (45)
For the set (37) [Vaananen] gave
dimℚ = p (46)
True also for the set (38).
q-world numbers
∞∑n=0
1
Fan+b/∈ ℚ (47)
∞∑n=0
1
Lan+b/∈ ℚ (48)
where a, b,∈ ℤ+, Fn and Ln are the Fibonacci and Lucas numbers,
respectively; F0 = 0,F1 = 1, L0 = 2, L1 = 1.
[Andre-Jeannin]: a = 1; [Bundschuh+Vaananen] with a measure.
[Prevost+T.M.]: a, b ≥ 1 with irrationality measures; [Merila].
IRRATIONALITY MEASURE
By an effective irrationality measure (exponent) of a given number
� ∈ ℂp we mean a number � = �(�) ≥ 2 which satisfies the
condition: for every � > 0 there exists an effectively computable
constant H0(�) ≥ 1 such that∣∣∣∣� − M
N
∣∣∣∣p
>1
H�+�(49)
for every M/N ∈ ℚ with H = max{∣M∣, ∣N∣} ≥ H0(�).
Irrationality measures of explicit numbers
�(e) = 2 Classical (50)
�(log 2) ≤ 3.8914 [Rukhadze] (51)
�(log 3) ≤ 5.125 [Salikhov] (52)
�(�) ≤ 8.0161 [Hata] (53)
�(�(2)) ≤ 5.4413 [Rhin+Viola] (54)
�(�(3)) ≤ 5.5139 [Rhin+Viola] (55)
Irrationality measures of explicit numbers
�1 =1
F1 +
1
F2 +
1
F3 + . . ., �(�1) = 2 (56)
�2 =1
L1 +
1
L2 +
1
L3 + . . ., �(�2) = 2 (57)
∣∣∣∣�i − M
N
∣∣∣∣ ≥ C
N2+D/√logN
(58)
Linear forms
Let Θ ∈ ℂp be a number to be studied.
a) p =∞. I an imaginary quadratic field and ℤI ring of integers.
b) p ∈ ℙ = {2, 3, 5, ...}. I = ℚ.
Linear forms
In the following theorems put
Q(n) = ea(n), R(n) = e−b(n) (59)
where
a(n) = an, b(n) = bn (classical) (60)
or
a(n) = an log n, b(n) = bn log n (classical) (61)
and
a(n) = an2, b(n) = bn2 (q-world). (62)
Linear forms
Assume that
Rn = BnΘ− An ∀n ∈ ℕ (63)
are numerical approximation forms satisfying
Bn, An ∈ ℤI (64)
BnAn+1 − AnBn+1 ∕= 0, (65)
∣Bn∣ ≤ Q(n), and also (66)
∣An∣ ≤ Q(n), if p ∕=∞ (67)
∣Rn∣p ≤ R(n) (68)
for all n ≥ n0 with some positive a and b and a < b, if p ∕=∞.
Linear forms/Axiomatic
Let the above assumptions be valid. Then for every � > 0 there
exists a constant H0 = H0(�) ≥ 1 such that∣∣∣∣Θ− M
N
∣∣∣∣p
> H−�−� (69)
for all M,N ∈ ℤI with H ≥ H0, where (by folkflore)
� = 1 + a/b, H = ∣N∣, if p =∞, (70)
� =b
b − a, H = max{∣M∣, ∣N∣}, if p ∈ ℙ. (71)
Kalle Leppala (Master thesis): Axiomatic for more general a(n)
and b(n).
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.
Pade approximations/Classical case
First we will study the classical series F (t) with it’s derivativies
ΔbF (t), where Δ = t ddt .
Denote d = max{degP(y), degQ(y)} and let d ,m ∈ ℤ+ and the
numbers �1, ..., �m be given.
We start by giving explicit type II Pade approximations for the
series
ΔbF (t�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, 93–148 (1927)]
Maier’s product formula
Let l ,m ∈ ℤ+ and � = t(�1, ..., �m) be given and define
�i = �i (l , �) bym∏t=1
(�t − w)l =ml∑i=0
�iwi . (73)
Thenml∑i=0
�i ik�i
t = 0 (74)
for all t ∈ {1, ...,m}; k ∈ {0, ..., l − 1}.
Maier’s product formula
Moreover
�i = (−1)i∑
i1+...+im=i
(l
i1
)⋅ ⋅ ⋅(
l
im
)⋅ �l−i1
1 ⋅ ⋅ ⋅�l−imm . (75)
Pade approximations/Classical case
Let b, d , l ,m, � ∈ ℕ, b < d and choose m numbers �1, ..., �m. Put
Bl ,�(t) =ml∑i=0
tml−i�i (l , �)[Q]i+�+⌊l/d⌋−1
[P]i+�. (76)
Then
Bl ,�(t)ΔbF (�j t)− Al ,�,b,j(t) = Rl ,�,b,j(t), (77)
where
degt Bl ,�(t) = ml , degt Al ,�,b,j(t) ≤ ml + �− 1 (78)
ordt=0
Rl ,�,b,j(t) ≥ ml + ⌊l/d⌋+ �. (79)
Pade approximations/Classical case
Thus we have a gap of lenght ⌊l/d⌋ in the power series expansion
Bl ,�(t)ΔbF (�j t) = Al ,�,b,j(t) + Rl ,�,b,j(t). (80)
The polynomials Bl ,�(t) are Pade approximant denominators in
variable t for the functions Fb,j(t) = ΔbF (t�j),
b = 0, 1, ..., d − 1; j = 1, ...,m.
Also we say that (77–79) define a Pade approximation with the
degree and order parameters
[degt B, degt A ≤, ordt=0
R ≥] = [ml ,ml + �− 1,ml + ⌊l/d⌋+ �]
(81)