Irrationality proof of certain Lambert series using little q-Jacobi polynomials

Journal of Computational and Applied Mathematics 233 (2009) 680–690

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Irrationality proof of certain Lambert series using little q-Jacobi polynomials J. Coussement, C. Smet ∗ Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium

article

abstract

info

Article history: Received 1 February 2008

We apply the Padé technique to find rational approximations to h± (q1 , q2 ) =

Keywords: Padé approximation Little q-Jacobi polynomials Irrationality Measure of irrationality q-series

∞ X

qk1

k=1

1 ± qk2

,

0 < q1 , q2 < 1, q1 ∈ Q, q2 = 1/p2 , p2 ∈ N \ {1}.

A separate section is dedicated to the special case qi = qri , ri ∈ N, q = 1/p, p ∈ N \ {1}. In this construction we make use of little q-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h± (q1 , q2 ) and give an upper bound for the irrationality measure. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In this paper we investigate quantities of the form h± (q1 , q2 ) :=

∞ X

qk1

k=1

1 ± qk2

,

0 < q1 , q2 < 1, q1 ∈ Q, q2 = 1/p2 , p2 ∈ N \ {1}.

(1.1)

Since we will assume q1 , q2 to be fixed, we will write h± = h± (q1 , q2 ). In the special case qi = qri , by writing (1 +

q = 1/p, p ∈ N \ {1}, ri ∈ N,

)

qk2 −1

=

lim(1 − q) h+ = q↑1

∞ X j =0

(1.2)

(− ) and changing the order of summation, we clearly have   1 r1 (−1)j = Ψ −1, 1, r1 + jr2 r2 r2

P∞

j =0

qk2 j

where Ψ is the Lerch transcendent, which is a generalization of the Hurwitz zeta function and the polylogarithm function. Some particular cases are h+ (q, q) = − lnq 2 and h+ (q, q2 ) = βq (1) which are q-extensions of − ln 2 and β(1) = π /4, P∞ respectively. In the same manner h− can be seen as a q-analogue of the (harmonic) series k=1 (r1 + kr2 )−1 . − In 1948 Erdős proved that h (q, q) = ζq (1) is irrational when q = 1/2, see [1]. Later, Peter Borwein [2,3] showed that ζq (1) and lnq 2 are irrational whenever q = 1/p with p an integer greater than 1. Other irrationality proofs were found in, e.g., [4–10]. To the best of our knowledge, the sharpest upper bounds for the irrationality measure of ζq (1), lnq 2 and βq (1) which are known in the literature until now, are 2.46497868 [11], 2.92832530 [12] and 6.50379809 [13] respectively. As can be seen in Table 1, for these 3 constants we only improve the upper bound for the irrationality measure in the case of βq (1) (case +, r2 = 2).

∗

Corresponding author. E-mail addresses: [email protected] (J. Coussement), [email protected] (C. Smet).

0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.02.036

J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

681

Table 1 Some values of the upper bound m± (r2 ) for the irrationality measure of h± , see (1.14). r2

m− (r2 )

m+ (r2 )

2π 2 π 2 −2

≈ 2.508284762

2π 2

5π 2 +8−

≈ 3.362953864

1 2

π 2 −4

3

32π 2 16π 2 −69

4

54π 2 27π 2 −125

5

3456π 2 1728π 2 −8005

5π 2 +8− 3 5π 2 +7− 2

15π 2 +24−

≈ 3.769124031

6

172800π 2 86400π 2 −414281

8

132300π 2 66150π 2 −327931

9

1881600π 2 940800π 2 −4664047 71442π 2 35721π 2 −180973

22(π 2 +7)(2π 2 +7) 2

20π 2 6(π 2 +8)(425π 2 +1704)

√ 1

300π 2 +450− 6 525π 2 +840−

≈ 4.018510114

√

315π 2 +504− 31

≈ 9.115392347

6(2π 2 +15)(122400π 2 +464131)

2(π 2 +8)(62475π 2 +250756) 1680π 2

√

√

≈ 7.169882132

240π 2 420π 2

2100π 2 +3220− 16

≈ 4.109498873

≈ 7.577046429

2(3π 2 +22)(2424π 2 +8959)

25π 2 +40− 3

≈ 4.018388861

≈ 5.658303107

≈ 6.037384338

(π 2 +8)(99π 2 +392) 2 √ 72π

√ 1

≈ 3.889740382

2

√ 12π

90π 2 +132− 12

≈ 4.015077389

≈ 3.171897628

3(π 2 +8)(31π 2 +120)

√ 4π 1

≈ 3.767042717

7

2

3(π 2 +8)(3π 2 +8) 4π 2

√ 1

≈ 3.552067296

300π 2 150π 2 −743

10

√ 4π

≈ 7.912912894

≈ 9.132668031

6(3π 2 +23)(4037600π 2 +15689991)

252π 2 3(π 2 +8)(136269π 2 +549256)

≈ 8.826251294

≈ 9.793912472

In [14] Matala-aho and Prévost also considered quantities of the form (1.1). However, not all the numbers we prove to be irrational are covered by their result. To prove this irrationality we use a well-known lemma, which expresses the fact that a rational number can be approximated to order 1 by rational numbers and to no higher order [15, Theorem 186]. Lemma 1.1. Let x be a real number. Suppose there exist integers an , bn (n ∈ N) such that: (i) bn x − an 6= 0 for all n ∈ N; (ii) limn→∞ (bn x − an ) = 0; then x is irrational. Proof. Suppose x is rational, so write x = a/b with a, b coprime. Then bn a − an b is a nonzero integer sequence that tends to zero, which is a contradiction.  In Section 2 we construct rational approximations to h± . In particular, we extend the Padé approximation technique applied in [8] to prove the irrationality of ζq (1) and lnq 2 and use little q-Jacobi polynomials (which are a generalization of the q-Legendre polynomials). Section 3 then mainly consists of calculating the asymptotic behaviour of the ‘error term’. Section 4 points out what improvements can be made in the special case (1.2). If we define

η :=

13

λ− :=

13

−

4

+

3 2π 2

−

6 + 5π 2

s

13π 2 + 6

4π 2

π2 + 6

,

η+ :=

5 3

+

2

π2

−

p

36 + 30π 2



1 12

+

1



3π 2

,

(1.3)

√ 4

+

3 2π 2

13π 4 + 84π 2 + 36

−

4π 2

,

λ+ :=

5 3

+

2

π2

+

p

36 + 30π 2

     2 2  Y X  $ 1 α  3  2   − 2  α +  2 2 −1 2  π $ j r2  j
g + (α) =

3

Y

π2 +

$ |r2 $ prime

π2

12

−

1 3π 2



,

(1.4)

if α ≥ 1, (1.5) if α < 1,

X 1 X φ(r2 /g ) g 2 Y Y j−1 $2 $2 + 2 2 2 2 $ − 1 (l,r )=1 l 2 j r2 $ |r $ − 1 j|g g |r2 ,g >1 2 2

g |r2 ,g >1 r2 /g odd

+

1

2

X φ(r2 /g ) g 2 Y

4



 max

4

r22

1



4

, α2

$ |r2 $ prime

r2 /g even

$ prime

j prime

Y j − 1 max(0, 1 − α)2 $2 + $ 2 − 1 j|g j 2

for even r2 ,

j prime

(1.6)

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J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

g + (α) =

X φ(r2 /g ) g 2 Y X 1 Y j−1 $2 $2 + π 2 $ |r2 $ 2 − 1 (l,r )=1 l2 g |r ,g >1 3 j r22 $ |r $ 2 − 1 j|g 2 2 2 j prime $ prime $ prime   X max(0, 1 − α)2 4 Y 1 α2 $2 + + 2 max , 2 2 2 π $ |r2 $ − 1 (l,r )=1 4l r2 2 2 $ prime     1 2  1 Y $  4 , α 1 − + 2 max  for odd r2 , π 4 r2 $ |r $ + 1 2 2

Y

(1.7)

$ prime

γ − := −α −

1 2

+ g − (α),

γ + := −α −

1 2

+ g + (α),

(1.8)

1

1 + g − (α), κ + := α + + g + (α), 2 2 then our main results are the following.

κ − := α +

(1.9)

Theorem 1.2. Let q2 = 1/p2 with p2 ∈ N \ {1} and q1 ∈ Q with 0 < q1 < 1. Then the number h± , defined as in (1.1), is ± irrational. Moreover, there exist integer sequences a± n , bn such that

2 η± ± ± 1/n lim b± ≤ p2 < 1 n h − an

(1.10)

± 1/n2 lim |b± ≤ pλ2 . n |

(1.11)

n→∞

and n→∞

Theorem 1.3. Let q = 1/p with p ∈ N \ {1}, qi = qri and pi = pri with ri ∈ N, i = 1, 2 and (r1 , r2 ) = 1. Then the number h± , ± defined as in (1.1), is irrational. Moreover, there exist integer sequences a± n , bn such that

2 γ ± (r ) ± ± 1/n lim b± ≤ p2 2 < 1 n h − an

(1.12)

n→∞

and κ ± (r2 )

1/n lim |b± ≤ p2 n | 2

n→∞

.

(1.13)

As a side result of the irrationality, we also obtain an upper bound for the irrationality measure (Liouville–Roth number, order of approximation) for h± . Recall that this measure is defined as



 a 1 µ(x) := inf t : x − > t +ε , ∀ε > 0, ∀a, b ∈ Z, b sufficiently large , b

b

see, e.g., [16]. It is known that all rational numbers have irrationality measure 1, whereas irrational numbers have s irrationality measure at least 2. Furthermore, if bn x − an 6= 0 for all n ∈ N, |bn x − an | = O (b− n ) with 0 < s < 1 and |bn | < |bn+1 | < |bn |1+o(1) , then the measure of irrationality satisfies 2 ≤ µ(x) ≤ 1 + 1/s, see [16, exercise 3, p. 376]. Note that by (1.10) and (1.11), respectively (1.12) and (1.13), we get the asymptotic behaviour

  ± ± b h − a± = O (b± )η± /λ± +ε , n

n

n

for all ε > 0, n → ∞,

and for the special case (1.2)

  ± ± b h − a± = O (b± )γ ± /κ ± +ε , n

n

n

for all ε > 0, n → ∞,

which then implies the following upper bound for µ(h± ). Corollary 1.4. Under the assumptions of Theorem 1.2 we have 2 ≤ µ(h± ) ≤ ν ± where ν − = 4.299865877 and ν + = m± (r2 ) ≤ ν ± where m± (r2 ) = 1 −

κ± . γ±

6π 2

√

3(π 2 +4)−2

36+30π 2

√ 4π 5π 2 +6−

2

13π 4 +84π 2 +36

≈

≈ 11.471299; under the assumptions of Theorem 1.3 we have 2 ≤ µ(h± ) ≤

(1.14)

J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

683

Remark 1.5. In fact, Theorem 1.3 can also be applied if gcd(r1 , r2 ) = ρ 6= 1. In that case just note that h± (q1 , q2 ) = r0 r0 h± (q0 1 , q0 2 ) with r10 = r1 /ρ , r20 = r2 /ρ and q0 = qρ . Moreover, easy calculations show that for an integer a,

∞ X q1 qa2 k=1


k

1 − qk2

−

∞ X

qk1

k=1

1 − qk2

and

∞ X q1 qa2


k

1 + qk2

k=1

+ (−1)a+1

∞ X

qk1

k=1

1 + qk2

are both rational, since they are both a finite sum of rational numbers. Hence the numbers h± (q1 qa2 , q2 ) have the same irrationality measures for any integer a as long as q1 qa2 < 1. As a consequence we can now restrict ourselves to parameters r1 , r2 for which 1 ≤ r1 ≤ r2

gcd(r1 , r2 ) = 1.

and

2. Rational approximation We first focus on the general case (1.1), the special case (1.2) will be treated in Section 4. We use the notation q1 = s1 /t1 with gcd(s1 , t1 ) = 1, and p1 = 1/q1 . 2.1. Padé approximation To prove the irrationality of h± we will apply Lemma 1.1. So we need a sequence of ‘good’ rational approximations. To find these we will perform the (well-known) idea of Padé approximation to the Markov function f (z ) :=

∞ X k=0

where logq x =

qk1 z−

log x log q

qk2

0

logq x

q1

2

z−x

dq2 x x

,

(2.1)

and the q-integration is defined as

1

Z

1

Z =

g (x) dq x := 0

∞ X

qk g (qk ).

(2.2)

k=0

So, we look for polynomials Pn and Qn of degree n such that Qn (z )f (z ) − Pn (z ) = O z −n−1 ,




z → ∞.

(2.3)

As is well known in the Padé approximation theory (see, e.g., [17]) the polynomials Qn then satisfy the orthogonality relations 1

Z

logq x

Qn (x) xm q1

2

0

x−1 dq2 x = 0,

m = 0, . . . , n − 1.

(2.4)

(bq; q)k = 0, (q; q)k

(2.5)

Little q-Jacobi polynomials satisfy ∞ X

pn (qk ; a, b|q)pm (qk ; a, b|q)(aq)k

k=0

m 6= n,

in which the q-Pochhammer symbols are defined as

(x; q)n =

n −1 Y

(1 − xqj ).

j =0

Hence Qn are little q-Jacobi polynomials with a particular set of parameters, namely Qn (z ) = pn (z ; q1 p2 , 1|q2 ), see, e.g., [18, Section 3.12]. Lemma 2.1. The polynomials Qn have the explicit expressions Qn (z ) = (−1)n

n X (−1)k k=0

=

n X k=0

k(k−1) (p1 ; p2 )n+k −nk k p2 2 z (p1 ; p2 )k (p2 ; p2 )k (p2 ; p2 )n−k

n(n−1) + k(k2+1) −nk 2

pn1−k p2

(−1)k

hni k

p2

(p1 ; p2 )n+k (q2 z ; q2 )k . (p1 ; p2 )n (p2 ; p2 )k

(2.6)

(2.7)

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J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

Proof. From, e.g., [18, Section 3.12], we know that the polynomials satisfying the orthogonality conditions (2.4) have the hypergeometric expression Qn (z ) = Cn2 φ1



pn2 , q1 qn2

 q ; q z , 2 2

q1

which is (2.6) if we choose Cn = (−1)n

(p1 ; p2 )n . (p2 ; p2 )n

This particular choice simplifies some arithmetic calculations further on. Next we apply the transformation formula [18, (0.6.24)] and find

(pn2 ; q2 )n 3 φ2 (q1 ; q2 )n

Qn (z ) = Cn

giving the expression (2.7).



pn2 , q1 qn2 , q2 z

 q ; q , 2 2

q2 , 0 

It is easily checked that the Pn are connected with the polynomials Qn by the formula Pn ( z ) =

Qn (z ) − Qn (x)

1

Z

z−x

0

logq x 2

q1

x −1 d q 2 x .

(2.8)

Indeed, then Qn (z )f (z ) − Pn (z ) =

1

Z

Qn (x) z−x

0

logq x 2

q1

x−1 dq2 x,

(2.9)

and the conditions (2.3) are fulfilled. Lemma 2.2. The polynomials Pn have the explicit formulae k(k+1)

−nk−j−1 j k−1 X z (p1 ; p2 )n+k p1 p2 2 Pn (z ) = (−1) (−1) k−j−1 (p1 ; p2 )k (p2 ; p2 )k (p2 ; p2 )n−k j=0 p1 p2 −1 k=0 n

=

n X

n X

n−k+j

p1

k

n(n−1) + k(k2+1) −nk 2

p2

(−1)k

hni k

k=0

p2

j +1 k X (p1 ; p2 )n+k −j (q2 z ; q2 )k−j (p2 ; p2 )j−1 . p2 (p1 ; p2 )n (p2 ; p2 )k j=1 (p1 ; p2 )j

(2.10)

(2.11)

Proof. First of all we mention that (for k ∈ N ∪ {0}) 1

Z

z k − xk z−x

0

logq x

q1

2

x −1 d q 2 x =

k−1 X j =0

zj

∞ X

(k−1−j)` `

q1 =

q2

`=0

k−1 X

zj k−j−1

j =0

1 − q1 q2

.

So, applying (2.6) to (2.8) we easily obtain (2.10). Next, observe that k X (q2 z ; q2 )k − (q2 x; q2 )k j j +1 =− q2 (q2 z ; q2 )k−j (q2 x; q2 )j−1 z−x j =1

(2.12)

which one can prove by induction. Moreover, using the q-binomial series [19, Section 10.2], [20, Section 1.3] ∞ X (a; q)n n =0

(q; q)n

xn =

(ax; q)∞ , (x; q)∞

|q| < 1, |x| < 1,

we get 1

Z

logq x

(q2 x; q2 )j−1 q1 0

2

x−1 dq2 x = (q2 ; q2 )j−1

∞ X (qj2 ; q2 )` ` (q ; q )j−1 q1 = 2 2 . (q2 ; q2 )` (q1 ; q2 )j `=0

Combining (2.8), (2.7), (2.12) and (2.13) we then finally establish (2.11).



(2.13)

J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

685

2.2. Rational approximants to h± Notice that by the definition (2.1) of f we have h± = ∓

q1 q2

f (∓p2 ).

Following the idea of Padé approximation we could try to approximate h± by the sequence of rational numbers ∓q1 Pn (∓p2 )/[q2 Qn (∓p2 )]. However, we prefer the evaluation of f at ∓pN2 with N = bα nc and α > 0 a real number for which we can later on make the optimal choice. h± =

N −1 X

qk1



1 ± qk2 k=1

∓

q1

N

f (∓pN2 ).

q2

(2.14)

In this way we can benefit from the fact that the finite sum on the right hand side of (2.14) already gives a good approximation for h± . Moreover, the approximation (2.3) is useful for z tending to infinity. Hence the evaluation at the point ∓pN2 makes ± ± more sense, especially since n and N will tend to infinity too. So, a ‘natural’ choice for rational approximations a± n /bn to h then has the following expressions. Definition 2.3. Define



NP −1 ± a± pN1 Qn (∓pN2 ) n := en

qk1 1±qk2

k=1

 ∓ pN2 Pn (∓pN2 ) ,

(2.15)

N ± N b± n := en p1 Qn (∓p2 ),

(2.16)

±

where en are factors such that these are integer sequences. The following lemma gives a possible choice for the factors e± n . It contains the cyclotomic polynomials

Φn (x) =

n  Y

x−e

2π ik n



.

(2.17)

k=1, (k,n)=1

Their degree is denoted by Euler’s totient function φ(n), being the number of positive integers ≤n that are coprime with n. It is well known [21, Section 4.8] that xn − 1 =

Y

Φd (x),

n=

X

φ(d),

(2.18)

d|n

d|n

and that every cyclotomic polynomial is monic, has integer coefficients and is irreducible over Q[x]. Lemma 2.4. By taking max(0,n−N )2 2

N +n e− p2 n = s1

N −1

(p1 ; p2 )n

Y

Φd (p2 )

(2.19)

d=1 max(0,n−N )2 2

N +n e+ p2 n = s1

(p1 ; p2 )n

n Y d=1, d≡1 mod 2

bnc

Φd d (p2 )

max(n,2(N −1))

Y

Φd (p2 ),

(2.20)

d=2, d≡2 mod 2

± the a± n and bn , defined as in (2.15) and (2.16), are integer sequences.

Proof. We use the expressions (2.7) for Qn and (2.11) for Pn . It is an easy counting exercise to see that the power of s1 is N −k sufficient and that no powers of p2 or t1 are needed in e± ; p2 )k−j n , if N ≥ n. However, if n > N, for k > N the factor (∓p2 (present in the second term of a± ) contains negative powers of p , which explains the need for extra powers of p in this case. 2 2 n The factor (p1 ; p2 )n is obviously needed to make the second term of a± n an integer. The only possible remaining denominators

x2j −1 , we conclude that we only need to consider xj −1 which cyclotomic polynomials appear to which power in the numerator and denominator of Qn pN2 and Pn pN2 . A careful j

originate from factors p2 ±1. Using (2.18) together with the fact that xj +1 =

(±

count yields that the factors proposed in (2.19) and (2.20) are sufficient and optimal.

)

(±

)



Remark 2.5. Using (2.10) in the above proof, one could expect a power of p2 in the denominator of pn2 Pn (±pn2 ), and this n2 /2

of the order of p2 . This would totally ruin the asymptotics in the next section. However, Maple calculations showed the absence of a power of p2 in the denominator. This is why we had to use an equivalent formula for Pn , which is given by (2.11).

686

J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

3. Irrationality of h± ± ± ± ± In this section we look at the error term b± n h − an , where an and bn are defined as in Definition 2.3 and (2.19)–(2.20). Using (2.14) and (2.9) one easily sees that it has the integral representation

h

i

± ± ± N N N N b± n h − an = ∓en p2 Qn (∓p2 )f (∓p2 ) − Pn (∓p2 )

Qn (x)

1

Z

N = e± n p2

pN2 ± x

0

logq x

q1

x−1 dq2 x.

2

(3.1)

We will show that this expression is different from zero for all n ∈ N and obtain its asymptotic behaviour. Here we study R± n :=

1

Z 0

Qn (x) pN2

logq x 2

q1

±x

x −1 d q 2 x

(3.2)

and e± n separately. 3.1. Asymptotic behaviour of R± n We will need the following very general lemma for sequences of polynomials with uniformly bounded zeros. This can be found in, e.g., [8, Lemma 3], but we include a short proof for completeness. Lemma 3.1. Let {πn }n∈N be a sequence of monic polynomials for which deg (πn ) = n and the zeros xj,n satisfy |xj,n | ≤ M, with M independent of n. Then 1/n lim πn cxn = |x|,



2




|x| > 1, c ∈ C.

n→∞

Proof. Since |x| > 1, for large n we easily get 0 ≤ |cxn | − M ≤ |cxn − xj,n | ≤ |cxn | + M ,

j = 1, . . . , n.

This implies

|cxn | − M


n



n ≤ πn cxn ≤ |cxn | + M

and

1/n  1/n 
1/n2 M M ≤ πn cxn ≤ |x| |c | + n . |x| |c | − n |x| |x| The lemma then follows by taking limits.



For R± n , defined as in (3.2), we have the following asymptotic result. Here we use a similar reasoning as in [8] for the irrationality of ζq (1). Lemma 3.2. Let Qn be the polynomials (2.6) satisfying the orthogonality relations (2.4). Then R± n is different from zero for all n and

1/n2 −α−1/2 lim R± = p2 . n

(3.3)

n→∞

Proof. First of all observe that Qn (∓pN2 ) R± n = ∓

1

Z

Qn (x)

Qn (∓pN2 ) − Qn (x)

∓

0

pN2

−x

logq x 2

q1

x −1 d q 2 x +

1

Z

Qn2 (x) pN2

0

±x

logq x 2

q1

x−1 dq2 x.

The first integral on the right hand side vanishes because of the orthogonality relations (2.4) for the polynomial Qn . Furthermore, note that 0 ≤ x ≤ 1 so that 0<

pN2

1

Z

1

+1

Qn2

logq x

(x) q1

2

0

x

−1

dq2 x ≤ Qn (∓

pN2

) Rn ≤

1

±

pN2

This already proves that R± n 6= 0. Next, from [18, (3.12.2)] we get 1

Z

Qn2 0

logq x

(x) q1

2

x

−1

dq2 x =

qn1 1 − q1 q2n 2



(q2 ; q2 )n (q1 ; q2 )n

2

.

−1

1

Z

logq x

Qn2 (x) q1 0

2

x−1 dq2 x.

(3.4)

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687

Applying this on (3.4), we easily establish

1/n = 1. lim Qn (∓pN2 ) R± n

2



(3.5)

n→∞

Now write Qn (x) = Kn Qˆ n (x) where Qˆ n is monic. From (2.6) we get that the leading coefficient Kn has the expression

(pn2 ; q2 )n (q1 qn2 ; q2 )n n q . (q1 ; q2 )n (q2 ; q2 )n 2 Qn Qn Since i=1 pi2−1 ≤ |(pn2 ; q2 )n | ≤ i=1 pi2 , this gives the asymptotic behaviour Kn =

1/2

lim |Kn |1/n = p2 . 2

(3.6)

n→∞

Since the Qn are orthogonal polynomials with respect to a positive measure on [0, 1], their zeros are all in [0, 1]. From Lemma 3.1 we then also get


1/n2



= pα2 .

lim Qˆ n ∓pN2

n→∞

(3.7)

Applying (3.6) and (3.7) to (3.5) then completes the proof.



3.2. Asymptotic behaviour of e± n Since we are interested in the asymptotic behaviour of e± n , which consists of a product of cyclotomic polynomials, we need to know the asymptotic behaviour of the sum of totient functions (since the degree of a cyclotomic polynomial is given by this totient function). Some interesting asymptotic properties are lim

n 1 X

n→∞ n2

φ(aj) =

j =0

3a

Y

π

$ |a

2

1 X

n→∞ n2

φ(aj + b) =

j =0

3a

Y

π

$ |a

2

n→∞

$2 , $2 − 1

1 X n2 j = 0

φ(2(aj + b)) =

4a

π2

Y $ |a, $ ≥3 $ prime

$2 , $2 − 1

where (a, b) = 1, see [22,23,14]. To obtain the asymptotic behaviour of n Y

bnc Φd d (p2 )

d=1,dodd

we first notice that this is a polynomial of degree j k n

n jnk X d=1,dodd

Using

P∞

1 j =1 j 2

lim

n→∞

d

=

j ∞ X X

φ(d) =

φ(d).

j=1 d=1,dodd

π2 6

n Y

and (3.9) we conclude that

b nd c

Φd

!1/n2 (p2 )

d=1,dodd

=

1 3

.

We obviously also have the asymptotic properties 1/2

lim ((p1 ; p2 )n )1/n = p2 , 2

n→∞

lim s1n+N

n→∞

(3.9)

$ prime

n

lim

(3.8)

$ prime

n

lim

$ , $ +1


1/n2

= 1.

This leads us to the following asymptotic behaviour of e± n .

(3.10)

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J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

Corollary 3.3. For e± n defined as in (2.19)–(2.20) we have 2

2 1 1/n2 + 3α2 + max(0,21−α) 2 , lim e− = p2 π n

5 1/n2 + max(12,4α 6 π lim e+ = p2 n

2)

+ max(0,21−α)

n→∞

n→∞

2

.

(3.11)

As a result we now have



2



2

−α+ 3α2 + max(0,21−α)

− − 1/n lim b− = p2 n h − an

2

π

n→∞

1

3 + + 1/n lim b+ = p2 n h − an

,



2



2 2 −α+ max(12,4α ) + max(0,21−α)

π

n→∞

.

(3.12)

3.3. Proof of Theorem 1.2 ± In the previous sections we defined integer sequences a± n and bn and managed to find the asymptotic behaviour of ± ± b± h − a . Putting these results together, we can now prove Theorem 1.2. n n ± Proof of Theorem 1.2. In Lemma 2.4 we made sure that a± n and bn , defined as in (2.15) and (2.16), are integer sequences. Note that by (3.6), (3.7) and (3.11) we then get 3α 2 2

1/n lim |b− = p2π n | 2

+α+1+ max(0,21−α)

n→∞

2

,

2 2 α+ 43 + max(12,4α ) + max(0,21−α)

1/n lim |b+ = p2 n | 2

π

n→∞

.

(3.13)

Putting together these results, we obtain

   − − − |b− n h − an | = O (bn ) 

2 3α 2 −α+ max(0,1−α) 2 π2 3α 2 +α+1+ max(0,1−α)2 2 π2

 +ε 

 , 

for all ε > 0, n → ∞

and

   + + + |b+ n h − an | = O (bn ) 

2 max(4α 2 ,1) −α+ 1 + max(0,21−α) 3 π2 2 max(4α 2 ,1) +α+ 4 + max(0,21−α) 3 π2

 +ε 

 , 

for all ε > 0, n → ∞.

Minimizing this exponent under the condition α > 0 yields the optimal values

√ 1

α =− + −

13π 4 + 84π 2 + 36 2(π 2 + 6)

2

≈ 0.954564

and 1

1 p

2

12

α+ = − +

36 + 30π 2 ≈ 1.018607.

These optimal choices for α ± yield the asymptotic results stated in Theorem 1.2 and this finishes the proof of this theorem. These results are optimal in the sense that other choices for α ± will result in worse upper bounds for the irrationality measure, so the result stated in Corollary 1.4 can only be achieved with this choice of α ± .  4. Improvements on the results in the special case (1.2) Throughout this section we consider the special case given by (1.2). The only difference with the general case is that we can (in some cases considerably) improve the factor e± n that is needed to make the approximation sequences into integer sequences. The following lemma gives the enhanced formula for e± n , and can be seen as an analogue of Lemma 2.4. To obtain its optimal expression, we had to carefully consider which cyclotomic polynomials should be present, whence the much more complicated expression. Lemma 4.1. Let I = {i ∈ N : i|hr2 + r1 (0 ≤ h ≤ n − 1) or i|hr2 (1 ≤ h ≤ max(n, N − 1))}, J1 = {i ∈ N : (i, r2 ) = 1, i odd, i|hr2 + r1 (0 ≤ h ≤ n − 1)}, J2 = {i ∈ N : (i, r2 ) = 1, i even, i|hr2 + r1 (0 ≤ h ≤ n − 1) or i|2hr2 (1 ≤ h ≤ N − 1)},

J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

689



 i J3 = i ∈ N : (i, r2 ) = g > 1, odd, i|hr2 (1 ≤ h ≤ n) , g   i J4 = i ∈ N : (i, r2 ) = g > 1, even, i|hr2 (1 ≤ h ≤ n) or i|2hr2 (1 ≤ h ≤ N − 1) . g

By taking max(n−N ,0)(n−N +1) 2

−N e− n = p1 p2

Y

Φi (p),

(4.1)

i∈I max(n−N ,0)(n−N +1) 2

−N e+ n = p1 p2

Y

Φi (p)

i∈J1

Y

Φi (p)

i∈J2

Y

ng Φi (p)b i c

i∈J3

Y

Φi (p),

(4.2)

i∈J4

± the a± n and bn , defined as in (2.15) and (2.16), are integer sequences.

Proof. The proof is analogous to the proof of Lemma 2.4. Due to the fact that p1 and p2 are both powers of an integer p, ± we can completely express a± n and bn in terms of powers of p and cyclotomic polynomials in p. Counting of the latter is a time-consuming job, where it is necessary (especially in the + case) to divide the indices i of the cyclotomic polynomials into four distinct sets.  4.1. Asymptotic behaviour of e± n The fact that in this special case (1.2), the factor e± n as given in (4.1) and (4.2) is already written as a product of factors p and cyclotomic polynomials in p, makes the task of finding its asymptotic behaviour more straightforward: we can again apply the asymptotic formulae (3.8)–(3.10). For the product in e+ n where the indices are in J4 , we need an additional asymptotic formula, which generalizes (3.10). Lemma 4.2. For a, b integers with a > b > 0, (a, b) = 1, we have lim

n 1 X

n→∞ n2

φ(g (aj + b)) =

j =0

3ag

π

2

Y $ |ag $ prime

Y $ −1 $2 . 2 $ − 1 $ |g $

(4.3)

$ prime

Proof. The asymptotic behaviour of j=0 φ(aj + b) is given by (3.9), but since for a prime p we have that φ(pa) = (p − 1)φ(a) if p - a and φ(pa) = pφ(a) if p|a, we can use induction arguments to obtain the desired formula. 

Pn

For the product in e+ n where the indices are in J3 , we need one more formula. Lemma 4.3. For a, b integers with a > b > 0, (a, b) = g, we have lim

n 1 X

n→∞ n2

j =0



bn aj + b



φ(aj + b) =

b2 2a

Y $ |a

$ prime

Y $ −1 $2 . 2 $ − 1 $ |g $

(4.4)

$ prime

Proof. The sum for which we want to know the asymptotic behaviour can be written as a double sum without coefficients, in the following way: bn

c  aj n  ∞ bX X X bn φ(aj + b) = φ(ak + b). aj + b j =0 j=1 k=0

Applying Lemma 4.2 to our inner sum, we obtain one infinite sum where the j-dependence is only in a factor 1/j2 . Using ∞ X 1 j=1

j2

=

π2 6

we obtain the desired asymptotic formula.



To obtain an expression for the asymptotic behaviour that is as simple as possible, we need the following lemma.

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J. Coussement, C. Smet / Journal of Computational and Applied Mathematics 233 (2009) 680–690

Lemma 4.4. For any integer r we have that

Y $ |r $ prime

( 2 $ 2 X φ(h) φ(r /h) = 3 $ 2 − 1 h|r h r 1 h odd

for r even, for r odd.

Proof. The statement of the lemma can easily be checked for prime numbers r and for r = 1. Induction arguments can be applied to complete the proof: one has to make a distinction into several cases, e.g. if the statement holds for odd r, one can show that it also holds for rp, with p an odd prime and p|r. The induction arguments hold for all the cases, hence the statement in the lemma is true.  These lemmas and the other asymptotic formulae (3.8)–(3.10) allow us to conclude that the asymptotic behaviour of e± n as given in (4.1)–(4.2) is indeed given by the following lemma. Lemma 4.5. g ± (α)

1/n lim (e± ≤ p2 n ) 2

n→∞

,

with g ± (α) given in (1.5)–(1.7). 4.2. Proof of Theorem 1.3 As in the proof of Theorem 1.2, the only task left is to put all the asymptotic results together. Notice that the γ ± and κ ± still depend on the parameter α . To obtain the optimal results (the first few of which are stated in Table 1), we need to find ± the α that minimizes the upper bound for the irrationality measure, given by 1 − γκ ± as stated in Corollary 1.4. In the − case αopt turns out to be 1 for any r2 , whereas in the + case αopt is less than 1 for some r2 and greater than 1 for the other r2 . Acknowledgements We would like to thank W. Van Assche for careful reading and useful discussions. This work was supported by INTAS project 03-51-6637, by FWO projects G.0184.02 and G.0455.05 and by OT/04/21 of K.U. Leuven. The first author has been a postdoctoral researcher at the K.U. Leuven (Belgium). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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