ON LINEAR FORMS WITH COEFFICIENTS IN ℕ ζ(1 + ℕ)

The detailed version, Part 6

L. A. Gutnik

In memory of Professor Alexander Buchstab

Content

§0. Foreword.

§1. Pizzicato.

§2. Metamorphosis.

§3. Repeating of the main theme.

§4. The first culmination.

§5. The second culmination.

§6. Some corrections to the previous parts.

§0. Foreword

As it have been already remarked in the foreword and in the section 3 of the paper [40], the situation in [22] is much simpler than in [34]-[37],[39],[43]. The point z = −1 is not a singular point of the differential equation, considered in [22], but the point z = 1 is a singular point of the differential equation, considered in [34]-[37],[39],[43], and the matrices B_{i}(z, ν) in (2.3.5) of the paper [35] are not defined at the point z = 1 Therefore we must use the passage to the limit z → 1. Many unpleasant things may occure by this passage to the limit. As it have been proved in [37], the six polynomials (3.2.10) are solutions of the difference equation (3.1.106) in the section 1 of [37]. But one of these six polynomials vanishes at the point z = 1 (precisely γ^{∗}(1; ν) = 0) and therefore we have only five linear independent solutions of the equation (3.1.106), which has sixth order. Where is one more solution hidden ? The answer was given in [43]. In [43] we proved that coefficient q_{0}(z, ν^{−1}) vanishes at the point z = 1 and therefore the specified hidden solution is a trivial solution

x(ν_{0}) = 1, x(ν) = 0, ν ∈ ν_{0} + ℕ,

where ν_{0} = [1/ε_{2}(F)] + 1, and we can not take it in the account, if we pass to the equation (29) in [43], which has the fifth order.

Typeset by AMS-TEX

Let us consider the following simple difference equation

y(ν + 1) = \frac{ν}{(z − 1)ν + 1}y(ν).

For each z ≠ 1 it is a linear difference equation of Poincaré type. But for z = 1 this eqution is not a linear difference equation of Poincaré type .

In the section 1 I prove a some general Theorem (below Lemma 6.1.1), which give sufficient conditions for lineare difference equation

(0.1)  ∑_{k=0}^{n} a_{k}(ν^{−1})y(ν + k) = 0,

where a_{k}(ν^{−1}) ∈ ℂ(ν), k = 0, …, n − 1, a_{n}(ν^{−1}) = 1, a_{0}(ν^{−1}) ≠ 0, would be a difference equation of a Poincaré type. These conditions may be formulatad as follows:

the space of solutions of equation (0.1) must have a basis

β_{j}(ν),

where j = 1, …, n, for which there exist C > 0 and M > 0 such that

|β_{j}(ν)| ≤ CM^{ν}

for all the sufficiently big ν. The necessity of these conditions was formulated by A.Poincaré for the general (i.e. not necessary a_{k}(ν^{−1}) ∈ ℂ(ν)) lineare difference equation of Poincaré type and and proved by E.B.Van Vleck [2]. As the followig simple example shows, the reqirement

a_{k}(ν^{−1}) ∈ ℂ(ν)

for k = 0, …, n − 1 is essential for the discussed Theorem:

let a_{0}(ν) = 4 + (−1)^{ν}2, a_{1}(ν) = 4 + (−1)^{ν}; then the equation

y(ν + 2) + a_{1}(ν) + a_{0}(ν)y(ν) = 0

is not difference equation of Poincaré type, but for any its solution y(ν) there exists a constante C > 0 such that |y(ν)| ≤ C3^{ν}.

In the end of the section 1 I apply the above general Theorem to the considered in these publications case and show that equation (29) in [43] is a linear difference equation of Poincar’e type.

ON LINEAR FORMS WITH COEFFICIENTS IN ℕζ(1 + ℕ) 3

Let us consider the following linear difference equation

y(ν + 1) = \frac{(z − 1)ν + 2}{(z − 1)ν + 1}y(ν)

For each z ≠ 1 its characteristical polynomial is equal to λ − 1, but for z = 1 its characteristical polynomial is equal to λ − 2.

Where is an exit from this devil’s morass? Suddenly I remarked a small footpath there and I go on this footpath in the next sections.

Acknowledgment. I express my deepest thanks

to Professor B.Z.Moroz,

to Professor I.I. Piatetski-Shapiro,

to Professor A.G.Aleksandrov,

to Professor P.Bundshuh

and to Professor S.G.Gindikin for the support of my work.

§1. Pizzicato

Lemma 6.1.1. Let ε_{0} > 0 and

(1.1)  a_{k}(ν^{−1}) ∈ ℂ(ν), k = 0, …, n − 1

from (0.1) are well defined for ν ∈ ℕ + [1/ε_{0}], and

(1.2)  a_{0}(ν^{−1}) ∈ ℂ(ν) ≠ 0

for ν ∈ ℕ + [1/ε_{0}], Let

(1.3)  β_{j}(ν),

where j = 1, …, n, is solution of the equation (0.1), and for ν ∈ ℕ + [1/ε_{0}] solutions (4) compose a basis of the linear space over ℂ of all the solutions of the equation (0.1). Let further there exists C > 0 and M > 0 such that

(1.4)  |β_{j}(ν)| ≤ CM^{ν}

for ν ∈ ℕ + [1/ε_{0}].

Then the rational functions (1.1) are well defined at the point ν^{−1} = 0 and therefore the equatin (0.1) is a difference equation of a Poincaré type.

Proof. Let the assertion of the Lemma is not true. Then

0 ∈ 𝔍 = {j ∈ {0, …, n − 1}: a_{j}(ν^{−1}) ∈ ℂ(ν^{−1}) ≠ 0_{ℂ(ν)}} ≠ ∅.

Let

a_{j}(ν^{−1}) = b_{j}(ν^{−1})ν^{λ_{j}}

where j ∈ 𝔍, b_{j}(ν^{−1}) ∈ ℂ(ν^{−1}), these b_{j}(ν^{−1}) are well defined at the point ν^{−1} = 0 and b_{j}(0) ≠ 0. Then the equation (1) may be rewritten in the form

(1.5)  y(ν + n) + ∑_{j∈𝔍} b_{j}(ν^{−1})ν^{λ_{j}}y(ν + j) = 0,

The equation (6) was studied by N.E.Nörlund [19]. Following to N.E.Nörlund we let y(ν) = (ν!)^{α}w(ν) with ν ∈ ℕ + [1/ε_{0}] and

(1.6)  α = sup{λ_{j}/(n − j): j ∈ 𝔍}.

If α ≤ 0 then assertion of the Lemma, of course, is true. Let

α > 0.

Then equation (1.5) turns into equation

(1.7)  w(ν + n) + ∑_{j∈𝔍} c_{j}(ν)w(ν + j) = 0,

where c_{j}(ν^{−1}) = b_{j}(ν^{−1})ν^{λ_{j}−(n−j)α} for j ∈ 𝔍 and ν ∈ ℕ + [1/ε_{0}]. Let

𝔍_{1} = {j ∈ 𝔍: λ_{j}/(n − j) = α}.

Then

(1.8)  lim_{ν→∞}(c_{j}(ν)) = b_{j}(0) ≠ 0

for j ∈ 𝔍_{1} and

(1.9)  lim_{ν→∞}(c_{j}(ν)) = 0

for j ∈ 𝔍⧹𝔍_{1}. In view of (1.8) and (1.9) the equation (1.7) is an equation of the Poincaré type with characteristic polynomial

(1.10)  η^{n} + ∑_{j∈𝔍_{1}} b_{j}(0)η^{j}.

According to (1.8), the polynomial (1.10) has at least one different from zero root, let η_{0} be one of the different from zero roots of the polynomial (1.10). According to the Perron’s theorem, the equation (1.7) has a solution w_{0}(ν) with the following property:

for any ε ∈ (0, 1) (for example for ε = 1/2) the inequality

|w_{0}(ν)| ≥ |η_{0}(1 − ε)|^{ν},

holds infinitely many times when ν → ∞. For the corresponding to w_{0}(ν) solution

y_{0}(ν) = w_{0}(ν)(ν!)^{α}

of the equation (6) the inequality

|y_{0}(ν)| ≥ (ν!)^{α}|η_{0}(1 − ε)|^{ν},

with α > 0 holds infinitely many times when ν → ∞. On the other hand the solution y_{0}(ν) is a ℂ-linear combinaion of the solutions (1.3); therefore there exist a constants C_{0} > 0 and M_{0} > 0 such that

|y_{0}(ν)| ≤ C_{0}(M_{0})^{ν}

for ν ∈ ℕ + [1/ε_{0}].■

Let us consider the difference equation (29) in [43] now, i.e. the following difference equation:

(2.1)  ∑_{j=0}^{4} q_{j}^{∧∗}(ν^{−1})y(ν + j) + y(ν + 5) = 0,

where q_{j}^{∧∗}(ν^{−1}) ∈ ℚ(ν^{−1}), if j = 0 , …, 4, are well defined for ν ∈ [1/ε_{5}] + ℕ with some ε_{5} > 0 and q_{0}^{∧∗}(ν^{−1}) ≠ 0, when ν ∈ [1/ε_{5}] + ℕ As it is shown in [43],the equation (2.1) has a solutions

(2.2)  y(ν) = α_{k}^{∗}(1; ν),

where k = 0, 1, 3, 4, 5 and ν ∈ ℕ + [1/ε_{5}], and the polynomials α_{k}^{∗}(z; ν) are defined in (2.21), (2.22), (2.24), (2.26) and (2.28) of the paper [35]; the Lemma 4.1.3 in the paper [39] shows that solutions (2.2) compose a basis of the space of the solutions of the equation (2.1).

Making use of (2.16) - (2.18), (2.21), (2.22), (2.24), (2.26) and (2.28) of the paper [35] it is easy to check the following inequalities:

|α_{0}^{∗}(1; ν)| ≤ 2^{11Δν}2Δν,

|α_{1}^{∗}(1; ν)| ≤ 2^{11Δν}12(Δν)^{2},
|α_{3}^{∗}(1; ν)| ≤ 2^{11Δν}100(Δν)^{4},
|α_{4}^{∗}(1; ν)| ≤ 2^{11Δν}132(Δν)^{4},
|α_{5}^{∗}(1; ν)| ≤ 2^{11Δν}168(Δν)^{4}.

Therefore according to the Lemma 6.1 equation (2.1) is an equation of Poincaré type. So, the equation (2.1) (the equation (29) in [43]) is an equation of Poincaré type. Our nearest aim is to study the characteristic polynomial of this equation.

§2. Metamorphosis

Let 𝔨 be a field of characteristic zero contained in an integral domain R with quotient field K. Let us consider a non-trivial differentiation u on K. Suppose that u is trivial on a subfield K_{0}, containing 𝔨, and that the ring R contains an eigenvector z of u, whose eigenvalue λ is not equal to 0 and lies K_{0}. Clearly, the element z is transcendental over 𝔨.

For f ∈ R, let 𝔪_{f} denote the operator of multiplication by f in R, so that

𝔪_{f}: a ↦ fa

for a ∈ R, and let

𝔐_{R} = {𝔪_{f}: f ∈ R}.

Since R is an integral domain, the ring 𝔐_{R} is isomorphic to R. Let 𝔐^{∨} denotes the ring of all the 𝔨−endomorphisms of the additive group of the 𝔨−algebra R. For any ϕ ∈ 𝔐^{∨} and ψ ∈ 𝔐^{∨}, let us denote by ϕ∘ψ the composition of the operators ϕ and ψ, so that (ϕ∘ψ)f = ϕ(ψf) for f ∈ R. The above differentiation u is a 𝔨−linear operator on R. Let R < u > stand for the subring of the ring 𝔐^{∨} generated by u and 𝔐_{R}; an element of the ring R < u > will be called a differential polynomial. Since

(2.1)  u ∘ 𝔪_{f} = 𝔪_{uf} + 𝔪_{f} ∘ u,

it follows that every element D of R < u > may be represented in the form

(2.2)  D = ∑_{k=0}^{n(D)} 𝔪_{𝔠_{k}(D)} ∘ u^{k}

with n(D) ∈ ℕ − 1, u^{0} = 𝔪_{1_{R}} and 𝔠_{k}(D) ∈ R for k = 0 , …, n(D). We may rewrite the equality (2.2) in the form

(2.3)  D = ∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)} ∘ u^{k},

where 𝔠_{k}(D) = 0 for k ∈ n(D) + ℕ. For n ∈ ℕ − 1, let R|_{n} < u > be the set of those elements D in R < u > with 𝔠_{k}(D) = 0 for k > n in (2.3). Clearly,

R < u >= ⋃_{n=0}^{∞} R|_{n} < u > .

Applying an operator D in R|_{n} < u > to 1, z , …, z^{n}, one can see that, for k ∈ ℕ−1, the element 𝔠_{k}(D) in (2.3) is uniquely defined and 𝔠_{k} is a 𝔨−linear map from the ring R < u > to R. For D ≠ 0, let

deg(D) := sup{k ∈ ℕ − 1: 𝔠_{k}(D) ≠ 0}

denote the degree of the differential polynomial D. Let

(2.4)  uD = ∑_{k=0}^{∞} 𝔪_{u𝔠_{k}(D)} ∘ v^{k},

then

(2.5)  u ∘ D = uD + D ∘ u = uD + ∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)} ∘ u^{k+1}

for D ∈ R < u > .

For μ ∈ ℚ let 𝔒^{(μ)} denote the ring those elements in ℚ(z, ν^{−1}), which are welldefined in the two points (1, μ) and (1, 0); the ring 𝔒^{(0)} will be simply denoted by 𝔒. Let 𝔅 be the ring of those elements in ℚ(z), which are well-defined at the point z = 1. The field ℚ(z) can be regarded as a subfield of the field ℚ(z, ν^{−1}), so that 𝔅 ⊂ 𝔒^{(μ)} for μ ∈ ℚ. Clearly, the ring 𝔒^{(μ)} is an integral domain. First let R = 𝔒^{(μ)}, then K = ℚ(z; ν^{−1}); let K_{0} = ℚ(ν^{−1}). For δ = z \frac{∂}{∂z} , let v = ν^{−1}δ, and we take u = v. Then the independent variable z will play the role of the eigenvector z, the corresponding eigenvalue λ being equal to ν^{−1}, and 𝔨 = ℚ; by definition, ν^{−1} ∈ K_{0}. Thereby we have constructed the rings 𝔒^{(μ)} < v > and, in particular, the ring 𝔒 < v >. We shall consider also the ring 𝔅 < μδ >, which we obtain, on letting R = 𝔅, K = ℚ(z), K_{0} = ℚ and taking u = μδ. In this case, the independent variable z will play the role of the eigenvector z, the corresponding eigenvalue λ being equal to μ, and 𝔨 = ℚ. Thus K_{0} = 𝔨 = ℚ in this case.

Let 𝔒^{∗} < v > stand for the set of those differential polynomials D in 𝔒 < v >, for which 𝔠_{0}(D)(1, 0) ≠ 0, and let, for n ∈ ℕ − 1,

𝔒|_{n}^{∗} < v >= 𝔒^{∗} < v > ∩ 𝔒|_{n} < v > .

For k ∈ ℕ − 1 and D ∈ 𝔒^{∗} < v >, let

𝔟_{k}(D) = 𝔠_{k}(D) + v𝔠_{k+1}(D) − (v𝔠_{0}(D))𝔠_{k+1}(D)/𝔠_{0}(D).

Then 𝔟_{k}(D) ∈ 𝔒 for k ∈ ℕ − 1 and D ∈ 𝔒^{∗} < v >,

(2.6)  𝔟_{k}(D) = 𝔠_{k}(D)

for k ∈ ℕ − 1 + n and D ∈ 𝔒|_{n}^{∗} < v >,

(2.7)  𝔟_{k}(D)(z, 0) = 𝔠_{k}(D)(z, 0)

for k ∈ ℕ − 1 and D ∈ 𝔒^{∗} < v >.

Let

𝔡: 𝔒^{∗} < v >→ 𝔒^{∗} < v >

denote the map defined by the following equality

𝔠_{k}(𝔡D) = 𝔟_{k}(D)

for all D ∈ 𝔒^{∗} < v > and k ∈ ℕ − 1. It follows from (2.6) and (2.7) that, for n ∈ ℕ − 1,

(2.8)  𝔡(𝔒|_{n}^{∗} < v >) ⊂ 𝔒|_{n}^{∗} < v > .

Lemma 6.2.1. If D ∈ 𝔒^{∗} < v >, then

(2.9)  (v − 𝔪_{v𝔠_{0}(D)/𝔠_{0}(D)}) ∘ D = (𝔡D) ∘ v.

Proof. It follows from (2.5) that

(v − 𝔪_{v𝔠_{0}(D)/𝔠_{0}(D)}) ∘ D = v ∘ D = D ∘ v + vD − 𝔪_{v𝔠_{0}(D)/𝔠_{0}(D)}) ∘ D =

∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)} ∘ v^{k+1}) + 𝔪_{v𝔠_{0}(D)} + ∑_{k=0}^{∞} 𝔪_{v𝔠_{k+1}(D)} ∘ v^{k+1} − ∑_{k=0}^{∞} 𝔪_{v𝔠_{0}(D)𝔠_{k}(D)/𝔠_{0}(D))} ∘ v^{k} =

∑_{k=0}^{∞}(𝔪_{𝔠_{k}(D)} + 𝔪_{v𝔠_{k+1}(D)} − 𝔪_{v𝔠_{0}(D)𝔠_{k+1}(D)/𝔠_{0}(D))}) ∘ v^{k+1}.

■ Since v𝔒 ⊂ ν^{−1}𝔒, it follows from (2.5) that

D ∘ v^{n} − v^{n} ∘ D ∈ ν^{−1}𝔒 < v >

for D ∈ 𝔒 < v >, n ∈ ℕ − 1 and that

D_{1} ∘ D_{2} − D_{2} ∘ D_{1} ∈ ν^{−1}𝔒 < v >

for D_{1} ∈ 𝔒 < v >, D_{2} ∈ 𝔒 < v >. Therefore, for each D ∈ 𝔒 < v >, the right ideal D ∘ 𝔒 < v > +ν^{−1}𝔒 < v > is a two-sided ideal; here

D ∘ 𝔒 < v >:= {D ∘ D_{1}: D_{1} ∈ 𝔒 < v >}.

In particular, it follows that the right ideal

𝔪_{(z−1)}𝔒 < v > + ν^{−1}𝔒 < v >

is a two-sided ideal.

Lemma 6.2.2. If D ∈ 𝔒^{∗} < v >, k ∈ ℕ − 1, then

(2.10)  v^{k}Ker(D) ⊂ Ker (𝔡^{k}D), 𝔡^{k}D − D ∈ ν^{−1}𝔒 < v >

Proof. We apply induction on k + 1. For k = 0 relations (2.10) are obvious. Let relations (2.10) are true for some k ∈ ℕ − 1 and let D_{1} = 𝔡^{k}D. In view of (2.8) and (2.9),

(v − 𝔪_{v𝔠_{0}(D_{1})/𝔠_{0}(D_{1})}) ∘ D_{1} = (𝔡D_{1}) ∘ v

and therefore

vKer(D_{1}) ⊂ Ker (𝔡D_{1}) = Ker (𝔡^{k+1}D).

Accoding to assumpion of induction, v^{k} Ker (D) ⊂ Ker (D_{1}). Therefore

v^{k+1} Ker (D) ⊂ v Ker (D_{1}) ⊂ Ker (𝔡^{k+1}D).

In wiew of (2.7), 𝔡D_{1} − D_{1} ∈ ν^{−1}𝔒 < v > . Accoding to assumpion of induction,

D_{1} − D ∈ ν^{−1}𝔒 < v > .

Therefore 𝔡^{k+1}D − D = 𝔡D_{1} − D ∈ ν^{−1}𝔒 < v > . So,the relaitions (2.10) are true also for k + 1. ■

Let us consider the projection 𝔯_{n} defined, for every n ∈ ℕ and D ∈ 𝔒 < v >, by the equality

𝔯_{n}(D) = D − 𝔪_{𝔠_{n}}(D) ∘ v^{n}.

It follows that

𝔯_{n}𝔒^{∗} < v > ⊂ 𝔒^{∗} < v > .

Let 𝔠_{n}(D) ∈ (z − 1)𝔒 for some n in ℕ and some D in 𝔒|_{n}^{∗} < v >, and let k ∈ ℕ − 1. Then it follows from (2.6) that

(2.11)  𝔡^{k}D − 𝔯_{n}(𝔡^{k}D) = 𝔪_{c_{n}(𝔡^{k}D)} ∘ v^{n} = 𝔪𝔠_{n}(D) ∘ v^{n} ∈ (z − 1)𝔒 < v >,

(2.12)  𝔯_{n}(𝔡^{k}D)Ker(𝔡^{k}D) ⊂ (z − 1)𝔒,

and it follows from (2.10) that

(2.13)  (𝔯_{n}(𝔡^{k}D) ∘ v^{k})Ker(D) ⊂ 𝔯_{n}(𝔡^{k}D)Ker(𝔡^{k}D) ⊂ (z − 1)𝔒.

If, besides, (𝔠_{n−1}(D))(1; 0) ≠ 0, then it follows from (2.7) that

(2.14)  𝔠_{n−1}(𝔡^{k}D)(1; 0) ≠ 0

for k ∈ ℕ − 1. Moreover, for each s ∈ ℕ − 1 there exists ε(D, s) > 0 such that the coefficients

𝔠_{i}(𝔡^{k}D)(z; ν^{−1}), 0 ≤ k ≤ s, 0 ≤ i ≤ n,

the coefficient

(𝔠_{0}(𝔡^{k}D)(z; ν^{−1}))^{−1},

and the coefficients

(𝔠_{n−1}(𝔡^{k}D)(z; ν^{−1}))^{−1}, 0 ≤ k ≤ s,

are well-defined on the set

{(z; ν^{−1}): |z − 1| < ε(D, s), |ν^{−1}| < ε(D, s)}.

Thus

𝔠_{i}(𝔡^{k}D) ∈ 𝔒^{(μ)}, (𝔠_{0}(𝔡^{k}D))^{−1} ∈ 𝔒^{(μ)}, (𝔠_{n−1}(𝔡^{k}D))^{−1} ∈ 𝔒^{(μ)}

for 0 ≤ k ≤ s, 0 ≤ i ≤ n and μ ∈ [0, ε(D, s)).

Let θ_{μ}: 𝔒^{(μ)} → 𝔅 be the homomorphism defined by the equality

(θ_{μ}f)(z) = f(z; μ)

for f ∈ 𝔒^{(μ)}. By a slight abuse of notation, we denote by θ_{μ} both the natural extension of this mapping to the set of the matrices with entries in 𝔒^{(μ)} and the homomorphism

θ_{μ}: 𝔒^{(μ)} < v >→ 𝔅 < μδ >

defined by the equality

θ_{μ}(D) = ∑_{i=0}^{∞} 𝔪_{θ_{μ}(𝔠_{i}(D))}(μδ)^{i}

for D ∈ 𝔒^{(μ)} < v > (it is clear that θ_{μ}(D) ∈ 𝔅 < μδ > for D ∈ 𝔒^{(μ)} < v >). The homomorphism θ_{μ} defines a structure of an unitary module over 𝔒^{(μ)} < v > on the ring 𝔅.

Let 𝔯_{μ,n} be the projection defined, for every μ ∈ ℚ, n ∈ ℕ and D ∈ 𝔅 < μδ >, by the equality 𝔯_{μ,n}(D) = D − 𝔪_{𝔠_{n}}(D) ∘ (μδ)^{n}. Then 𝔯_{n}(𝔅 < μδ >) ⊂ 𝔅 < μδ > for μ ∈ ℚ, n ∈ ℕ, and the diagram

(2.15)  \begin{matrix}𝔒^{(μ)} < v > & ⟶^{r_{n}} & 𝔒^{(μ)} < v > \\ θ^{(μ)}↓ & & ↓θ^{(μ)} \\ 𝔅 < μδ > & ⟶_{r_{μ,n}} & 𝔅 < μδ > \end{matrix}

is commutative.

Let 𝔅^{∗} < μδ > denote the set of those differential polynomials D in 𝔅 < μδ > for which 𝔠_{0}(D)(1) ≠ 0, and let

𝔅^{∗}|_{n} < μδ >:= 𝔅^{∗} < μδ > ∩ 𝔅|_{n} < μδ >

for n ∈ ℕ − 1. Let D ∈ 𝔅^{∗} < μδ >; since c_{k}(D) ∈ 𝔅, 𝔅 ⊂ 𝔒, and c_{0}(D)(1, 0) = c_{0}(D)(1) ≠ 0, for each k ∈ ℕ − 1 there is an element

𝔟_{k}(D) = 𝔠_{k}(D) + v𝔠_{k+1}(D) − (v𝔠_{0}(D))𝔠_{k+1}(D)/𝔠_{0}(D)

in 𝔒^{μ} as above. Let

𝔟_{μ,k}(D) = θ_{μ}(𝔟_{k}(D)),

then

b_{μ,k}(D) = 𝔠_{k}(D) + μδ𝔠_{k+1}(D) − (μδ𝔠_{0}(D))𝔠_{k+1}(D)/𝔠_{0}(D) ∈ 𝔅

for k ∈ ℕ − 1. Let, further,

𝔡_{μ}: 𝔅^{∗} < μδ >→ 𝔅 < μδ >

be the mapping defined by the equality

(2.16)  𝔠_{k}(𝔡_{μ}D) = 𝔟_{μ,k}(D) = θ_{μ}(𝔟_{k}(D))

for D ∈ 𝔅^{∗} < μδ > and k ∈ ℕ − 1. Since

𝔅 ⊂ ⋂_{μ∈ℚ} 𝔒^{(μ)}

and θ_{μ} is identical on 𝔅, it follows that homomorphism θ_{μ} is surjective. Let

𝔒^{(μ∗)} < v >= {D ∈ 𝔒 < v >: 𝔠_{0}(D)|_{z=1,(ν)^{−1}=mu} ≠ 0}.

Clearly

(2.17)  θ_{μ}(O^{(μ∗)} < v >) ⊂ 𝔅^{∗} < μδ > .

Lemma 6.2.3 The diagram

(2.18.)  \begin{matrix}𝔒^{(μ∗)} < v > & ⟶^{𝔡} & 𝔒^{(μ)} < v > \\ θ^{(μ)}↓ & & ↓θ^{(μ)} \\ 𝔅 < μδ > & ⟶_{𝔡_{μ}} & 𝔅 < μδ > \end{matrix}

is commutative.

Proof. Let

D ∈ 𝔒^{(μ∗)} < v > .

Then 𝔡(D) ∈ 𝔒^{(μ)} < v > and

θ_{μ}(𝔡(D)) ∈ 𝔅 < μδ >

It follows from (2.9) that 𝔡_{μ}(θ_{μ}(D)) is defined, belongs to B^{∗} < μδ > and

(2.19)  𝔡_{μ}(θ_{μ}(D)) = θ_{μ}(𝔡(D)).

■ Lemma 6.2.4. Suppose that

D ∈ 𝔒^{∗} < v >, s ∈ ℕ − 1, 0 ≤ k ≤ s,

μ ∈ [0, ε(D, s)). Then (𝔡_{μ})^{k}(θ_{μ}(D)) is defined, belongs to B^{∗} < μδ > and

(2.20)  (𝔡_{μ})^{k}(θ_{μ}(D)) = θ_{μ}(𝔡^{k}(D)).

Proof. We apply induction on k + 1. For k + 1 = 1 the assertion of the Lemma is obvious. Let s ∈ ℕ, 0 ≤ k < s and assertion of the Lemma is true for k Accoding to above definition of ε(D, s),

(2.21)  D_{1} = 𝔡^{k}(D) ∈ 𝔒^{(μ∗)} < v > .

According to the Lemma 6.2.3, 𝔡_{μ}(θ_{μ}(D_{1})) is defined, belongs to B^{∗} < μδ > and

(2.22)  𝔡_{μ}(θ_{μ}(D_{1})) = θ_{μ}(𝔡(D_{1})) = θ_{μ}(𝔡^{k+1}(D)).

According to asumption of induction (𝔡_{μ})^{k}(θ_{μ}(D)) is defined, belongs to 𝔅^{∗} < μδ > and

(𝔡_{μ})^{k}(θ_{μ}(D)) = θ_{μ}(𝔡^{k}(D)).

Therefore 𝔡_{μ}((𝔡_{μ})^{k}(θ_{μ}(D))) is defined, belongs to 𝔅 < μδ > and

(2.23)  𝔡_{μ}((𝔡_{μ})^{k}(θ_{μ}(D))) = 𝔡_{μ}(θ_{μ}(𝔡^{k}(D)) = 𝔡_{μ}(θ_{μ}(D_{1}))

So, (𝔡_{μ})^{k+1}(θ_{μ}(D)) is defined, belongs to 𝔅 < μδ > and, according (2.22) and (2.23)

(2.24)  (𝔡_{μ})^{k+1}(θ_{μ}(D)) = θ_{μ}(𝔡^{k+1}(D)).

Therefore

(2.25)  𝔠_{0}((𝔡_{μ})^{k+1}(θ_{μ}(D))) = 𝔠_{0}(θ_{μ}(𝔡^{k+1}(D))).

According to above definition of ε(D, s), and to inequality k +1 ≤ s, it follows from (2.25) that 𝔠_{0}((𝔡_{μ})^{k+1}(θ_{μ}(D)))|_{z=1} ≠ 0 and therefore (𝔡_{μ})^{k+1}(θ_{μ}(D)) is defined, belongs not only to 𝔅 < μδ > but also to 𝔅^{∗} < μδ > ■

Lemma 6.2.5. Let are fulfilled the conditions of the Lemma 6.2.4. Let further

D ∈ 𝔒_{n}^{∗} < v >, 𝔠_{n−1}(D))(1; 0) ≠ 0, 𝔠_{n}(D) ∈ (z − 1)𝔒.

Let further s ∈ ℕ − 1, 0 ≤ k ≤ s μ ∈ [0, ε(D, s)), and 0 ≤ i ≤ n. Then

(2.26)  𝔠_{i}(𝔡^{k}D) ∈ 𝔒^{(μ)}, (𝔠_{0}(𝔡^{k}D))^{−1} ∈ 𝔒^{(μ)}

(2.27)  (𝔠_{n−1}(𝔡^{k}D))^{−1} ∈ 𝔒^{(μ)},

(2.28)  (θ_{μ}(v) − 𝔪_{θ_{μ}(v𝔠_{0}(D)/𝔠_{0}(D))}) ∘ θ_{μ}(D) = (𝔡_{μ}θ_{μ}(D)) ∘ θ_{μ}(v),

(2.29)  (θ_{μ}(v))^{k}Ker(θ_{μ}(D)) ⊂ Ker ((𝔡_{μ})^{k}θ_{μ}(D)),

(2.30)  (𝔡_{μ})^{k}θ_{μ}(D) − 𝔯_{n,μ}((𝔡_{μ})^{k}θ_{μ}(D)) ∈ (z − 1)𝔒^{(μ)},

(2.31)  𝔯_{n,μ}((𝔡_{μ})^{k}θ_{μ}(D))Ker((𝔡_{μ})^{k}θ_{μ}(D)) ⊂ (z − 1)𝔅,

(2.32)  (𝔯_{n,μ}((𝔡_{μ})^{k}θ_{μ}(D)) ∘ θ_{μ}(v^{k})) Ker (θ_{μ}(D)) ⊂
𝔯_{n,μ}((𝔡_{μ})^{k}θ_{μ}(D)) Ker ((𝔡_{μ})^{k}θ_{μ}(D)) ⊂ (z − 1)𝔅

for 0 ≤ k ≤ s and μ ∈ [0, ε(D, s)).

Proof. The includings (2.26) directly follow from the Lemma 6.2.3. The including (2.27) directly follows from the above definition of ε(D, s). It follows from (2.9) that

(2.33)  (θ_{μ}(v) − 𝔪_{θ_{μ}(v𝔠_{0}(D)/𝔠_{0}(D))})θ_{μ}(∘D) = θ_{μ}((𝔡D)) ∘ θ_{μ}(v).

¿From (2.33) and (2.20) follows (2.28). To prove (2.29) we use induction on k. for For k = 0 the including (2.29) is obvious. Let k < s and including (2.25) is true for k. Let D_{1} = 𝔡^{k}D. In view of (2.8), (2.21) and (2.28),

(2.34)  (θ_{μ}(v) − 𝔪_{θ_{μ}(v𝔠_{0}(D_{1})/𝔠_{0}(D))}) ∘ θ_{μ}(D_{1}) = (𝔡_{μ}θ_{μ}(D_{1})) ∘ θ_{μ}(v),

It follows from (2.34) and (2.20) that

(2.35)  θ_{μ}(v)Ker(θ_{μ}(D_{1})) ⊂ Ker (𝔡_{μ}θ_{μ}(D_{1}))

In view of (2.35) and (2.20)

(2.36)  θ_{μ}(v)Ker(𝔡_{μ}^{k}θ_{μ}(D)) ⊂ Ker (𝔡_{μ}^{k+1}θ_{μ}(D))

According to induction assumption,

(2.37)  (θ_{μ}(v))^{k}Ker(θ_{μ}(D)) ⊂ Ker ((𝔡_{μ})^{k}θ_{μ}(D)),

and therefore

(2.38)  (θ_{μ}(v))^{k+1}Ker(θ_{μ}(D)) ⊂ θ_{μ}(v) Ker ((𝔡_{μ})^{k}θ_{μ}(D)),

It follows from (2.37) and (2.38) that (2.29) is true for k + 1. So, (2.29) is proved. Clearly, θ_{μ}((z − 1)𝔒^{(μ)} < v >)) = (z − 1)𝔅(μδ). In view of (2.11)

(2.39)  θ_{μ}(𝔡^{k}D) − θ_{μ}(𝔯_{n}(𝔡^{k}D)) = θ_{μ}(𝔪_{c_{n}(𝔡^{k}D)} ∘ v^{n}) =
θ_{μ}(𝔪_{𝔠_{n}(D)} ∘ v^{n}) ∈ (z − 1)𝔅 < μδ >

Applying to (2.39) the equality (2.22) and making use of the commuativity of the diagram (2.15), we obtain (2.30). So, (2.30) is proved. The including (2.31) directly follows from (2.30). Finally, (2.32) directly follows from (2.29) and (2.31). ■

Let are fulfilled the conditions of the Lemma 6.2.5. Then, in view of (2.32),

(θ_{μ}(𝔯_{n,μ}(𝔡^{k}D)))((θ_{μ}(v^{k}))f) ∈ (z − 1)𝔅

and

0 = (θ_{μ}((𝔠_{0}(𝔡^{k}D))(1; μ))(θ_{μ}(v^{k+0})f)(1) + … +

(2.40)  (θ_{μ}((𝔠_{n−1}(𝔡^{k}D))(1; μ))(θ_{μ}(v^{k+n−1})f)(1)

for f(z) ∈ ℚ[z] ∩ Ker (θ_{μ}(D)). Let n ≥ 2. In view of (2.27), let B_{k}^{∨}(D; z; ν^{−1}) denote the (n − 1) × (n − 1)− matrix, which has

−(𝔠_{j−1}(𝔡^{k}D))(z; ν^{−1})/(𝔠_{n−1}(𝔡^{k}D))(z; ν^{−1})

on the intersection of the last row with the j-th column for 1 ≤ j ≤ n − 1, has, in the case n > 2, the unit matrix on the intersection of the first n − 2 rows with the the last n − 2 columns and, if n > 2, has 0 in first n − 2 rows of the first column. For k ∈ ℕ − 1 let

C_{k}^{∨}(D; z; ν^{−1}) = ∏_{i=1}^{k} B_{k−i}^{∨}(D; z; ν^{−1}).

It follows from (2.7) that, for k ∈ ℕ − 1,

(2.41)  B_{k}^{∨}(D; z; 0) = B_{0}^{∨}(D; z; 0) and (B_{0}^{∨}(D; z; 0))^{k} = C_{k}^{∨}(D; z; 0);

therefore, for s ∈ ℕ, 0 ≤ k ≤ s, and ν^{−1} ∈ [0, ε(D, s)), if μ = ν^{−1}, then

{B_{k}^{∨}(D; z; ν^{−1}), C_{k}^{∨}(D; z; ν^{−1})} ⊂ Mat_{n−1}(𝔒^{(μ)}).

For T = T (z) ∈ ℂ[z]) and m ∈ ℕ we denote by means X_{m,T} = X_{m,T} (z; ν^{−1}) the column in (ℂ[z; ν^{−1}])^{m}, whose i-th element, where i ∈ [1, m]∩ℕ, is equal to v^{i−1}T. Given H = 𝔠_{0}(H)v^{0} + … + 𝔠_{s}(H)v^{s} in 𝔒 < v >, let

F (H; D; z; ν^{−1}) = 𝔠_{0}(H)C_{0}^{∨}(D; z; ν^{−1}) + … + 𝔠_{s}(H)C_{s}^{∨}(D; z; ν^{−1});

since C_{k}^{∨}(D; z; 0) = (B_{0}^{∨}(D; 1; 0))^{k} for k ∈ ℕ − 1, it follows that

(2.42) F (H; D; 1; 0) = 𝔠_{0}(H)(1; 0)(B_{0}^{∨}(D; 1; 0))^{0}+ … +𝔠_{s}(H)(1; 0)(B_{0}^{∨}(D; 1; 0))^{s}.

If H ∈ 𝔒^{(μ)} < v > and f(z) ∈ ℚ[z] ∩ Ker(θ_{μ}(D)), then, in view of (2.40)

θ_{μ}(B_{k}^{∨}(D; z; ν^{−1})v^{k}X_{n−1,f})(1) = θ_{μ}(v^{k+1}X_{n−1,f})(1),
θ_{μ}(C_{k}^{∨}(D; z; ν^{−1})X_{n−1,f})(1) = θ_{μ}(v^{k}X_{n−1,f})(1),
(θ_{μ}(H)X_{n−1,f})(1) = (F (H; D; 1; μ)X_{n−1,f})(1)

for 0 ≤ k ≤ s.

Suppose that i ∈ {0, 1}. Let us define the differential polynomials

D_{i} = b_{i,0}^{∨}(z, ν^{−1})v^{0} + … + b_{i,5}^{∨}(z, ν^{−1})v^{5} + (z − 1)v^{6}

and

P_{i} = p_{i,0}(ν^{−1})v^{0} + … + p_{i,3Δ+3i}(ν^{−1})v^{3Δ+3i},

corresponding to the polynomials D_{i}^{∗}(z, ν^{−1}, w) and P_{i}^{∗}(ν^{−1}, w) in (2.3.3)-(2.3.4) of the paper [36]. We have

𝔠_{6}(D_{i}) ∈ (z − 1)𝔒, (𝔠_{5}(D_{i}))(1; 0) ≠ 0, D_{i} ∈ 𝔒_{6}^{∗} < v >,

and

P_{i} ∈ 𝔒_{3Δ+3} < v > ∩𝔒^{(μ)} < v >

for μ ∈ ℚ. Let us take, in the formulae of this section,

D = D_{i}, H = P_{i}, n = 6, s = 3Δ + 3i, f = f_{1}^{∗}(z; ν + i),

and let

ε_{13} = min(ε(D_{0}, 3Δ), ε(D_{1}, 3Δ + 3)).

Suppose, moreover, that ν ∈ ℕ and ν^{−1} < ε_{13}. Then

(θ_{ν^{−1}}(P_{i})X_{5,f_{1}^{∗}(z;ν+i)})(1) = (F (P_{i}; D_{i}; 1; ν^{−1})X_{5,f_{1}^{∗}(z;ν+i)})(1).

Since the ring Q[ν^{−1}] < v > is commutative and P_{i} ∈ (Q[ν^{−1}] < v >, it follows that

P_{i}X_{5,f_{1}^{∗}(z;ν+i)} = X_{5,P_{i}f_{1}^{∗}(z;ν+i)}

and, in of (2.3.6) of the paper [36],

(F (P_{0}; D_{0}; 1; ν^{−1})X_{5,f_{1}^{∗}(z;ν)})(1) = (F (P_{1}; D_{1}; 1; ν^{−1})X_{5,f_{1}^{∗}(z;ν+1)})(1).

§3. The repeating of the main theme

We can apply Lemmata 2.4.5 and 2.4.4. of the paper [36] now. If η runs through the set of all the different from 1 roots of the polynomial D^{∧}(1, η) defined in the equality (2.4.56) of the paper [36], then the function w = (ηd_{2} + d_{1})/(η − 1) ranges over all the roots of the polynomial D^{∼}(1, w) defined in (2.4.3) of the paper [36]. Therefore there exist ε_{14} > 0 and q_{j}^{∧}(1, ν^{−1}) ∈ ℌ({1}, ε_{14}) with j ∈ [0, 4] ∩ ℕ such that y(ν) = f_{1}^{∗}(1; ν) satisfies for ν ∈ ℕ + [1/ε_{14}], the equation of the Poincaré type

(3.1)  y(ν + 5) + q_{4}^{∧}(1, ν^{−1})y(ν + 4) + … + q_{0}^{∧}(1, ν^{−1})y(ν + 0) = 0,

for ν ∈ ℕ + [1/ε_{1}4]. The polynomial

(3.2)  F_{1}^{∨}(η) = (η − h^{∼}(γ_{1})) ∏_{k=0}^{1} ∏_{l=0}^{1}(η − h^{∼}(η_{k}^{∧}(1, (2l − 1)π/3, δ_{0})))

is the characteristic polynomial of equation (3.1).

§4. The first culmination.

Let now R = ℚ(ν^{−1}), where ν^{−1} is an independent variable. As above, in the section 3, let 𝔪_{f}: a ↦ fa, a ∈ R, denotes the operator of multiplication by the element f ∈ R. Since R is a commutative ring R with a unit element 1_{R}, it follows that the ring 𝔐_{R} = {𝔪_{f}: f ∈ R} of all the R− endomorphisms of R is isomorphic to R. Let 𝔐^{∨} denotes the ring the ℚ−endomorphisms of the additive group of the ℚ−algebra R. As above, for ϕ ∈ 𝔐^{∨} and ψ ∈ 𝔐^{∨}, the element ϕ ∘ ψ in 𝔐^{∨} is defined by the formula (ϕ ∘ ψ)f = ϕ(ψf) for f ∈ R. The action of ℚ−linear operator ▽: R → R on R = ℚ(ν^{−1}) is given by the formula

(4.1)  ▽: f(ν^{−1}) ↦ f (\frac{ν^{−1}}{ν^{−1} + 1}) , (▽)^{−1}: f(ν^{−1}) ↦ f (\frac{ν^{−1}}{−ν^{−1} + 1})

for f(ν^{−1}) ∈ R; clearly, the operator ▽ turns ν into ν + 1 and the operator (▽)^{−1} turns ν into ν − 1. Let R < ▽ > stand for the subring of the ring 𝔐^{∨}, generated by ▽ and 𝔐_{R}. Since

(4.2)  ▽ ∘ 𝔪_{f} = 𝔪_{▽f} ∘ ▽,

an element D of the ring R < ▽ > can be represented in the form

(4.3)  D = ∑_{k=0}^{n(D)} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k},

with n(D) ∈ ℕ − 1, ▽^{0} = 𝔪_{1_{R}} and 𝔠_{k}(D) ∈ R for all the numbers k ∈ [0, n(D)] ∩ ℕ; let 𝔠_{k}(D) = 0 for k ∈ n(D) + ℕ, then

(4.4)  D = ∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k}.

For n ∈ ℕ − 1, let R|_{n} < ▽ > denotes the set of those elements D ∈ R < ▽ >, which can represented in the form (4.4) with 𝔠_{k}(D) = 0 if k > n or, equivalently, in the form

(4.5)  D = ∑_{k=0}^{n} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k}.

Clearly, R < ▽ > may be regarded as the subring of the ring 𝔐^{∨} generated by the ring 𝔐_{R} and the operator △ := ▽ − 𝔪_{1_{R}}. An element D of the set R|_{n} < ▽ > can be represented in the form

(4.6)  D = ∑_{k=0}^{n} 𝔪_{𝔞_{k}(D)} ∘ △^{k},

where △^{0} = 𝔪_{1_{R}} and 𝔞_{k}(D) ∈ R for 0 ≤ k ≤ n. On applying the operator D to the elements 1, ν , …, ν^{n}, one concludes that both the coefficients 𝔞_{k}(D), 0 ≤ k ≤ n,

in the equality (4.6) and the coefficient 𝔠_{k}(D) = ∑_{κ=k}^{n} \binom{κ}{k}𝔞_{κ}(D), 0 ≤ k ≤ n, in (4.4)

define ℚ−linear mappings

𝔞_{k} : R|_{n} < ▽ >→ R, 𝔠_{k} : R|_{n} < ▽ >→ R

respectively, and, clearly, those mappings are uniquely defined. An element of the ring R < ▽ > will be called a difference polynomial. For D ≠ 0_{R<▽>}, let

deg(D) := sup{k ∈ ℕ − 1: 𝔠_{k}(D) ≠ 0}.

For any D ∈ R < ▽ >, let us define a difference polynomial ▽D by means of the equality

▽D = ∑_{k=0}^{∞} 𝔪_{▽𝔠_{k}(D)} ∘ ▽^{k}.

It follows that

▽ ∘ D = ▽D ∘ ▽ = ∑_{k=0}^{∞} 𝔪_{▽c_{k}(D)} ∘ ▽^{k+1}

for any D ∈ R < u >. Let 𝔒 denotes the ring consisting of all the elements of the field R = ℚ(ν^{−1}), which are well-defined in the point ν^{−1} = 0. Let 𝔒 < ▽ > be the ring of difference polynomials P ∈ R < ▽ > with 𝔠_{k}(P) ∈ 𝔒 for k ∈ ℕ − 1. For n ∈ ℕ − 1, let

𝔒|_{n} < ▽ >:= 𝔒 < ▽ > ∩ R_{n} < ▽ > .

It is clear that the mappings f ↦ ▽f and f ↦ (▽)^{−1}f are automorphisms of the ring 𝔒. Therefore 𝔒 < ▽ > is a subring of the ring R < ▽ > .

Lemma 6.4.1. Suppose that

A ∈ R < ▽ >, B ∈ R < ▽ >, deg(A) = m ≥ deg(B) = n.

Then there exist the unique D_{1} ∈ R < ▽ > and A_{1} ∈ R < ▽ > such that

(4.7)  deg(D_{1}) = m − n, D_{1} − 𝔠_{m−n}(D_{1}^{∨}) ∘ (▽)^{m−n} = 0_{R<▽>},

and or A_{1}^{∨} = A−D_{1} ∘B is equl to 0_{R<▽>}, or deg(A_{1}) < deg(A). If c_{m}(B) = 1_{ℚ(ν)}, then 𝔠_{m−n}(D_{1}) = 𝔠_{m}(A). Moreover,

c_{m}(B) = 1_{ℚ(ν)}, A ∈ 𝔒 < ▽ >, B ∈ 𝔒 < ▽ >

then A_{1} ∈ 𝔒 < ▽ > and D_{1} ∈ 𝔒 < ▽ > .

Proof. Clearly,

D_{1} = 𝔪_{𝔠_{m}(A)((▽)^{m−n}c_{m}(B))^{−1}} ∘ (▽)^{m−n}.

■ Lemma 6.4.2. Suppose that

A ∈ R < ▽ >, B ∈ R < ▽ >, deg(A) = m ≥ deg(B) = n.

Then there exist the unique D_{0} ∈ R < ▽ > and A_{0} ∈ R < ▽ > such that

deg(D_{0}) = m − n

and or A_{0} = A − D_{0}B is equl to 0_{R<▽>,} or deg(A_{0}) < deg(B). If c_{m}(B) = 1_{ℚ(ν)}, then 𝔠_{m−n}(D_{0}) = 𝔠_{m}(A). Moreover, if

c_{m}(B) = 1_{ℚ(ν)}, A ∈ 𝔒 < ▽ >, B ∈ 𝔒 < ▽ >

then A_{0} ∈ 𝔒 < ▽ > and D_{0} ∈ 𝔒 < ▽ > .

Proof. We apply an induction on k = m − n + 1. If k = m − n + 1 = 1, then the assertion of the Lemma coincides with the asserion of the Lemma 6.4.1. Let further k = m − n + 1 > 1 and assertion of the Lemma is true for all κ ∈ ℕ, which are less than k First we shall prove existence. According to the Lemma 6.4.1, there exist the unique D_{1} ∈ R < ▽ >, and A_{1} ∈ R < ▽ > such that deg(D_{1}) = m − n and A_{1} = A − D_{1}B or is equl to 0_{R<▽>}, or deg(A_{1}) < deg(A). If c_{m}(B) = 1_{ℚ(ν)}, then 𝔠_{m−n}(D_{1}) = 𝔠_{m}(A). Moreover,

c_{m}(B) = 1_{ℚ(ν)}, A ∈ 𝔒 < ▽ >, B ∈ 𝔒 < ▽ >

then A_{1} ∈ 𝔒 < ▽ > and D_{1} ∈ 𝔒 < ▽ > . If A_{1} = 0_{R<▽>}, or deg(A_{1}) < deg(B), then we can take D_{0} = D_{1} and A_{0} = A_{1}. Let deg(A_{1}) ≥ deg(B). Then

1 ≤ κ = deg(A_{1}) − deg(B) + 1 < deg(A) − deg(B) + 1 = m − n + 1.

But the induction assumtion implies the existence of the unique D_{0}^{∨} ∈ R < ▽ > and A_{0} ∈ R < ▽ > such that deg(D_{0}^{∨}) = deg(A_{1})−deg(B) and or A_{0}^{∨} = A_{1} −D_{0}^{∨}B is equl to 0_{R<▽>}, or deg(A_{0}) < deg(B). Moreover,

c_{m}(B) = 1_{ℚ(ν)}, A_{1} ∈ 𝔒 < ▽ >, B ∈ 𝔒 < ▽ >

then A_{0} ∈ 𝔒 < ▽ > and D_{0}^{∨} ∈ 𝔒 < ▽ > . We can take

(4.8)  D_{0} = D_{1} + D_{0}^{∨}

now. We prove uniqueness now. Let a desired D_{0} is represented in the form (4.8), where

D_{1} = 𝔠_{m−n}(D) ∘ (▽)^{m−n}

and D_{0}^{∨} = D_{0} − D_{1} or is equl to 0_{R<▽>}, or deg(D_{0}^{∨}) < deg(D_{0}). Then D_{1} satisfies to the condition (4.7) and A_{1} = A − D_{1} ∘ B = D_{0}^{∨} ∘ B + A_{0}. Clearly, or A_{1} is equl to 0_{R<▽>,} or deg(A_{1}) < deg(A). According to the Lemma 6.4.1, D_{1} is is defined uniquely. Therefore A_{1} is defined uniquely. Consequently, D_{0}^{∨} is defined uniquely. ■

Lemma 6.4.3. The ring R < ▽ > is the ring of principal left (and also right) ideals.

Proof. Let 𝔄 is some left nonzero ideal of the ring R < ▽ > . Then 𝔄 contains a nonzero element B, which has

(4.9)  deg(B) = inf{deg(C): C ∈ 𝔄, C ≠ 0_{R<▽>}} = n.

Let A ∈ 𝔄⧹{0_{R<▽>}}, m = deg(A). According to the Lemma 6.4.2, there exist the unique D_{0} ∈ R < ▽ > and A_{0} ∈ R < ▽ > such that deg(D_{0}) = m − n and or the difference polynomial A_{0} = A − D_{0}B is equl to 0_{R<▽>}, or deg(A_{0}) < deg(B). Clearly, A_{0} ∈ 𝔄. The inequality deg(A_{0}) < deg(B) contradicts to (4.9). ■

Lemma 6.4.4. Suppose that

(4.10)  B ∈ R|_{m} < ▽ >, Q ∈ R|_{s} < ▽ >, 𝔠_{m}(B) = 𝔠_{s}(Q) = 1,

and

(4.11)  Q ∘ B ∈ 𝔒 < ▽ >,

then B ∈ 𝔒 < ▽ > and Q ∈ 𝔒 < ▽ > .

The first proof. Clearly,

(4.12)  Ker (B) ∈ Ker (Q ∘ B).

According to the E.B.Van Vleck’s theorem for any y ∈ Ker (Q ∘ B) there exist the real numbers C > 0, M > 0 and ν_{0} ∈ ℕ such that

|y_{0}(ν)| ≤ C(M)^{ν}

for all the ν ∈ ν_{0} + ℕ. In view of (4.10) and according to the Lemma 6.1.1, we have inclusion B ∈ 𝔒 < ▽ > . According to the Lemma 6.4.2, Q ∈ 𝔒 < ▽ > . ■

The second (pure algebraically) Proof. This proof is based on the same idea, as well known Gauß’s Lemma, which they used for the proof that the ring of the polynomials of several variables over arbirary field is a factorial ring. Clearly, the ring 𝔒 is a local ring with the (single) prime ideal 𝔭 = (ν)^{−1}𝔒. Since, in view of (4.1), ▽𝔭 ⊂ 𝔭, (▽)^{−1}𝔭 ⊂ 𝔭, it follows that ν^{−1}𝔒 < ▽ > is a two-sided ideal of the ring 𝔒 < ▽ > and ▽ determines an automorphism of the factor-ring 𝔒/𝔭𝔒; this automorphism we will also denote by ▽. Let α be a natural epimorphism of the ring 𝔒 < ▽ > on the factor-ring 𝔎 = 𝔒 < ▽ > /ν^{−1}𝔒 < ▽ > . Clearly, the map a ↦ 𝔪_{a}, where a ∈ 𝔒, defines an isomorphism α^{∨} of the ring 𝔒 onto the ring

𝔒^{∨} = 𝔐_{𝔒} = {𝔪_{f}: f ∈ 𝔒}.

Let 𝔭^{∨} = α^{∨}(𝔭). Then restriction of α on 𝔒^{∨} defines a natural epimorphism of the ring 𝔒^{∨} on the factor-ring 𝔎_{0} = 𝔒^{∨}/𝔭^{∨}; we denote this restriction also by α. Clearly, if D ∈ 𝔒 < ▽ > is represented in the form (4.4) with 𝔠_{k}(D) ∈ 𝔒, then

α(D) = ∑_{k=0}^{∞} α(𝔪_{𝔠_{k}(D)})(α(▽))^{k}.

If α(D) ≠ 0_{𝔎}, then let deg(α(D)) = sup{k ∈ ℕ − 1, α(m_{𝔠_{k}(D)}) ≠ 0_{𝔎_{0}}}. Cleary, when D_{k} ∈ 𝔒|_{n} < ▽ >, α(D_{k}) ≠ 0_{𝔎}, where k = 1, 2, then, in view of (4.2),

(4.13)  deg(α(D_{1} ∘ D_{2})) = deg(α(D_{1})) + deg(α(D_{2}))

and therefore α(D_{1} ∘ D_{2})) ≠ 0_{𝔎}. If D ∈ 𝔒 < ▽ >, is represented in the form (4.4) with 𝔠_{k}(D) ∈ 𝔒, and γ ∈ ℤ, then then D ∘ 𝔪_{ν^{γ}} = m_{ν^{γ}} D_{1} where

D_{1} = ∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)(1+kν^{−1})^{γ}} ▽^{k} ∈ 𝔒 < ▽ >,

(4.14)  α(D_{1}) = ∑_{k=0}^{∞} α(𝔪_{𝔠_{k}(D)(1+kν^{−1})^{γ}})(α(▽))^{k} ∈ 𝔒 < ▽ >,

the equalityα(D) = 0_{𝔎} holds if and only if α(D_{1}) = 0_{𝔎}, moreover, if α(D) ≠ 0_{𝔎}, then

(4.15)  deg(α(D)) = deg(α(D_{1})).

Let now

(4.16)  B = 𝔪_{ν^{β}} ∘ D, Q = 𝔪_{ν^{γ}} ∘ D_{2},

where D ∈ 𝔒|_{n} < ▽ >, D_{2} ∈ 𝔒|_{n} < ▽ >, α(D) ≠ 0_{𝔎}, α(D_{2}) ≠ 0_{𝔎}. Then in view of (4.11),(4,13) - (4.15), B ∘ Q = 𝔪_{ν^{β+γ}} ∘ D_{1} ∘ D_{2} ∈ 𝔒 < ▽ > . In view of (4.10) and (4.16) β ≥ 0, γ ≥ 0. And, in view of (4.11) and (4.15), β + γ = 0. ■

In accordance with (2.1), we consider D^{∧∗} ∈ R|_{5} < ▽ >, which has 𝔠_{5}(D^{∧∗}) = 1 and 𝔠_{j}(D^{∧∗}) = q_{j}^{∧∗}(ν^{−1}) for 0 ≤ j ≤ 4. Let, in accordance with (3.1), D^{∧} denotes the element of R|_{5} < ▽ > with 𝔠_{5}(D^{∧∗}) = 1 and 𝔠_{j}(D^{∧∗}) = q_{j}^{∧}(ν^{−1}) for 0 ≤ j ≤ 4. Then the equations (2.1) and (3.1), for a sufficiently small ε > 0 and ν ∈ [1/ε] + ℕ, have f_{1}^{∗}(1; ν) as their common solution. Let us consider the left ideal

R < ▽ > D^{∧∗} + R < ▽ > D^{∧}.

According to the Lemma 6.4.3. there exist

B ∈ R < ▽ >, A^{∧∗} ∈ R < ▽ >, A^{∧} ∈ R < ▽ >, Q^{∧∗} ∈ R < ▽ >, Q^{∧} ∈ R < ▽ >,

such that

(4.17)  𝔠(B)_{deg B} = 1_{R},

(4.18)  D^{∧∗} = Q^{∧∗} ∘ B, D^{∧} = Q^{∧} ∘ B,

(4.19)  A^{∧∗} ∘ D^{∧∗} + A^{∧∗} ∘ D^{∧∗} = B,

For a sufficiently small ε > 0 the fractions

𝔠_{k}(D^{∧∗}), 𝔠_{k}(A^{∧∗}), 𝔠_{k}(Q^{∧∗}), 𝔠_{k}(D^{∧}), 𝔠_{k}(A^{∧}), 𝔠_{k}(Q^{∧}), 𝔠_{k}(B),

are well defined for ν ∈ [1/ε]+ℕ and k ∈ ℕ−1. Let n = deg(B) and let us consider the cooresponding to B difference equation

(4.20)  ∑_{k=0}^{n} 𝔪_{𝔠_{k}(B)}y(ν + k) = 0.

Since the equations (2.1) and (3.1), for a sufficiently small ε > 0 and ν ∈ [1/ε] + ℕ, have f_{1}^{∗}(1; ν) as their common solution it follows from (4.19) that equation (4.20) for a sufficiently small ε > 0 and ν ∈ [1/ε] + ℕ, have f_{1}^{∗}(1; ν) as its solution. Therefore

(4.21)  n = deg(B) > 0

In view of (4.17)-(4-18) and according to the Lemma 6.4.4,

(4.22)  B ∈ 𝔒 < ▽ >, Q^{∧∗} ∈ 𝔒 < ▽ >, Q^{∧} ∈ 𝔒 < ▽ >,

(4.23)  𝔠(Q^{∧∗})_{deg(Q^{∧∗})} = 1_{R}, 𝔠(Q^{∧})_{deg(Q^{∧})} = 1_{R},

For each D ∈ 𝔒 < ▽ >, let P (D, η) denote the polynomial

(4.24)  D = ∑_{k=0}^{∞} 𝔠_{k}(D)|_{ν^{−1}=0}η^{k}.

In view of (4.2),

P (▽ ∘ D ∘ (▽)^{−1}, η) = P (D, η).

Therefore the map

D ↦ P (D, η)

is a homomomorphism (of course, an epimorphism) of the ring 𝔒 < ▽ > onto ℚ[η]. Consequently, in view of (4.18) we obtain the equality

(4.25)  P (D^{∧}, η) = P (Q^{∧}, η)P (B, η),

(4.26)  P (D^{∧∗}, η) = P (Q^{∧∗}, η)P (B, η).

In view of (4.17),

(4.27)  deg(P (B, z)) = deg(B) > 0

But the polynomial P (D^{∧}, η) coincides with the polyomial (3.2). In view of (3.1.28) in [37]

(4.28)  h^{∼}(η) = (η^{2} − 1)^{3}(1 − δ_{0})^{−3d_{1}}2^{−6}η^{3d_{1}}

(4.29)  |h^{∼}(η_{1}^{∧}(1, π, δ_{0})| = |h^{∼}(γ_{1})| = (δ_{0})^{3}/(1 + (δ_{0}))^{3Δ+3} < 1

But f_{1}^{∗}(1; ν) ∈ ℤ for ν ∈ ℕ − 1, and in view of the Lemma 4.1.4 in [39],

(4.30)  lim sup_{ν→∞} |f_{1}^{∗}(1; ν)|^{1/ν} ≥ 1,

In view of (4.29)-(4.30) and according to the Perron’s Theorem,

lim sup_{ν→∞} |f_{1}^{∗}(1; ν)|^{1/ν} = |h^{∼}(η)|,

where η is one of the different from γ_{1} roots of the polynomial (3.2), i.e. one of the roots of the polynomial F_{0}(η) in the Lemma 3.1.6 of the paper [37]. But according to the Lemma 3.1.6 of the paper [37], this F_{0}(η) is irreducible over ℚ. Therefore, in view of (4.25) and (4.27), the polynomial F_{0}(η) is a divisor of the polynomial P (B, η). In view of (4.26), we have proved the following important result:

Lemma 6.4.5. The polynomial F_{0}(η) in the Lemma 3.1.6 of the paper [37] is a divisor of the characteristical polynomial P (D^{∧∗}, η) of the equation (3.2)

Therefore

(4.31)  P (D^{∧∗}, η) = F_{0}(η)(η − λ_{0}),

where λ_{0} ∈ ℚ. Sice the polynomial F_{0}(η) is irreducible over ℚ, it follows that the multiplicity of the root λ_{0} ∈ ℚ of the polynomial P (D^{∧∗}, η) is equal to 1.

§5. The second culmination.

According (3.1.61) in [37], if r = 1, |ψ| ≤ π, then s = cos(ψ/2)

Lemma 6.5.1. Let

δ_{0} > 0, ψ^{∗} = 2 arccos(δ_{0}/4),

ρ_{0}(δ_{0}) = |\frac{η_{1}(1^{w}edge, ψ^{∗}, δ_{0})}{η_{1}^{∧}(1, π, δ_{0})}|^{1−δ_{0})} |\frac{η_{1}^{w}edge(1, ψ^{∗}, δ_{0}) + 1}{η_{1}^{∧}(1, π, δ_{0}) + 1}|^{δ_{0}} ×

|\frac{η_{1}^{w}edge(1, ψ^{∗}, δ_{0}) − 1}{η_{1}^{w}edge(1, π, δ_{0}) − 1}|^{δ_{0}} .

If δ_{0} ≤ 1/7, then

(5.1)  ρ_{0}(δ_{0}) < 1.

Proof. According to (3.1.71) in [37],

q_{0}(r, ψ^{∗}, δ_{0}) = \frac{4(δ_{0})^{2}}{(1 + δ_{0})^{2}} .

According to the Lemmata 3.1.2 and 3.1.4 in [37],

(5.2)  p_{0}(1, ψ^{∗}, δ_{0}) = 2|η_{0}^{∧}(1, ψ^{∗}, δ_{0}) − 1|^{2} = 2|η_{1}^{∧}(1, ψ^{∗}, δ_{0}) − 1|^{2} =

= 2\sqrt{q_{0}(1, ψ^{∗}, δ_{0})} = \frac{4δ_{0}}{1 + δ_{0}} .

In view of (3.1.41), (3.1.42) and (3.1.70) in [37],

(5.3)  p_{1}(1, ψ^{∗}, δ_{0}) = \frac{8}{(1 + δ_{0})^{2}} (\frac{(δ_{0})^{2}}{8} + \frac{1}{4} + |D_{0}(1, ψ^{∗}, δ_{0})|)

(5.4) p_{2}(1, ψ^{∗}, δ_{0}) = \frac{8}{(1 + δ_{0})^{2}} (1 − δ_{0} (\frac{(δ_{0})^{2}}{8} − 1) + \frac{(δ_{0})^{2}}{4} + |D_{0}(1, ψ^{∗}, δ_{0})|)

(5.5)  p_{0}(1, ψ^{∗}, δ_{0}) = \frac{8}{(1 + δ_{0})^{2}} ((2 + δ_{0})\frac{(δ_{0})^{2}}{8} + \frac{(δ_{0})^{2}}{4} + |D_{0}(1, ψ^{∗}, δ_{0})|)

Therefore

(5.6) p_{1}(1, ψ^{∗}, δ_{0}) − p_{0}(1, ψ^{∗}, δ_{0}) = \frac{8}{(1 + δ_{0})^{2}} (−(1 + δ_{0})\frac{(δ_{0})^{2}}{8} + \frac{1 − (δ_{0})^{2}}{4} ) =

\frac{2}{1 + δ_{0}} (1 − δ_{0} − \frac{(δ_{0})^{2}}{2} )

(5.7) p_{2}(1, ψ^{∗}, δ_{0}) − p_{0}(1, ψ^{∗}, δ_{0}) = \frac{8}{(1 + δ_{0})^{2}} (−(1 + δ_{0})\frac{(δ_{0})^{2}}{4} + (1 + δ_{0})) =

\frac{2}{1 + δ_{0}} (4 − (δ_{0})^{2})

In view of (3.1.43) in [37], (5.2), (5.6) and (5.7),

\frac{p_{1}(1, ψ^{∗}, δ_{0})}{2} = \frac{1}{1 + δ_{0}} (1 + δ_{0} − \frac{(δ_{0})^{2}}{2} ) ,

\frac{p_{2}(1, ψ^{∗}, δ_{0})}{2} = \frac{1}{1 + δ_{0}} (4 + 2δ_{0} − (δ_{0})^{2})),

\frac{p_{1}(1, ψ^{∗}, δ_{0})}{2} − \sqrt{q_{1}(1, ψ^{∗}, δ_{0})} = \frac{1}{1 + δ_{0}} (2δ_{0} − \frac{(δ_{0})^{2}}{2} );

\frac{p_{1}(1, ψ^{∗}, δ_{0})}{2} + \sqrt{q_{1}(1, ψ^{∗}, δ_{0})} = \frac{1}{1 + δ_{0}} (2 − \frac{(δ_{0})^{2}}{2} );

(\frac{p_{1}(1, ψ^{∗}, δ_{0})}{2})^{2} − q_{1}(1, ψ^{∗}, δ_{0}) = \frac{δ_{0}}{(1 + δ_{0})^{2}} (2 − \frac{(δ_{0})^{2}}{2} ) (2 − \frac{δ_{0}}{2} ) ≥

\frac{9δ_{0}}{4(1 + δ_{0})^{2}} ,

(5.8)  |η_{0}(1, ψ^{∗}, δ_{0})|^{2} ≥ \frac{1}{1 + δ_{0}} (1 + \frac{δ_{0}}{2} + \frac{3\sqrt{δ_{0}}}{2} )

\frac{p_{2}(1, ψ^{∗}, δ_{0})}{2} − \sqrt{q_{2}(1, ψ^{∗}, δ_{0})} = \frac{1}{1 + δ_{0}} (2δ_{0} − (δ_{0})^{2});

\frac{p_{2}(1, ψ^{∗}, δ_{0})}{2} + \sqrt{q_{2}(1, ψ^{∗}, δ_{0})} = \frac{1}{1 + δ_{0}} (8 + 2δ_{0} − (δ_{0})^{2});

(\frac{p_{2}(1, ψ^{∗}, δ_{0})}{2})^{2} − q_{2}(1, ψ^{∗}, δ_{0}) = \frac{δ_{0}}{(1 + δ_{0})^{2}} (2 − δ_{0})(8 + 2δ_{0} − (δ_{0})^{2}) ≥

\frac{9δ_{0}}{(1 + δ_{0})^{2}} ,

because for s = 1 − δ_{0} we have 0 < s < 1 and

(2 − δ_{0})(8 + 2δ_{0} − (δ_{0})^{2}) = (1 + s)(9 − s^{2}) = 9 + s(9 − s − s^{2}) > 9.

Further we have

(5.9)  |η_{0}(1, ψ^{∗}, δ_{0}) + 1|^{2} ≥ \frac{4}{1 + δ_{0}} (1 + \frac{δ_{0}}{4} + \frac{3}{2}\sqrt{δ_{0}}) .

In view of (3.1.56) , (3.1.43) in [37] and (5.2)

(5.10)  |η_{1}(1, π, δ_{0})|^{2} = q_{1}(r, ψ, δ_{0}), 4|η_{1}(1, π, δ_{0}) + 1|^{2} = q_{2}(1, ψ, δ_{0}),

\frac{|η_{1}(1, ψ^{∗}, δ_{0}) − 1|^{2}}{|η_{1}(1, π, δ_{0}) − 1|^{2}} = \frac{1 + δ_{0}}{2δ_{0}}

In view of (5.8) and (5.10),

(5.11)  \frac{|η_{1}(1, ψ^{∗}, δ_{0})|^{2}}{|η_{1}(1, π, δ_{0})|^{2}} = \frac{1}{|η_{0}(1, ψ^{∗}, δ_{0})|^{2}} ≤

\frac{1 + δ_{0}}{1 + \frac{δ_{0}}{2} + \frac{3\sqrt{δ_{0}}}{2}} .

In view of (5.9) and (5.10),

(5.12)  \frac{|η_{1}(1, ψ^{∗}, δ_{0}) + 1|^{2}}{|η_{1}(1, π, δ_{0}) + 1|^{2}} = \frac{4}{|η_{0}(1, ψ^{∗}, δ_{0}) + 1|^{2}} ≤ \frac{1 + δ_{0}}{1 + \frac{δ_{0}}{4} + \frac{3\sqrt{δ_{0}}}{2}} .

Therefore, in view of (5.10) - (5.12)

(ρ_{0}(δ_{0}))^{2} ≤ (\frac{1}{2δ_{0}} )^{δ_{0}} \frac{(1 + δ_{0})^{1+δ_{0}}}{1 + \frac{3\sqrt{δ_{0}}}{2}} ,

2 ln(ρ_{0}(δ_{0})) ≤ −δ_{0} ln(δ_{0}) − δ_{0}ln(2) + δ_{0} + (δ_{0})^{2} − \frac{3\sqrt{δ_{0}}}{2} + \frac{9δ_{0}}{8} <

(δ_{0}(ln(1/δ_{0}) − \sqrt{1/δ_{0}}) + δ_{0} + (δ_{0})^{2} − \sqrt{δ_{0}}/2 < 0,

because x + x^{3} < 1/2, if 0 ≤ x ≤ 2/5, ln(t) − \sqrt{(}t) decreases, if t > 4 and ln(6, 25) − \sqrt{6, 25} < 0. ■

Corollary 1. If δ_{0} ≤ 1/7 and s = cos(ψ/2) > δ_{0}/4 then

(5.13)  |\frac{η_{1}^{∧}(1, ψ, δ_{0})}{η_{1}^{∧}(1, π, δ_{0})}|^{1−δ_{0}} |\frac{η_{1}^{∧}(1, ψ, δ_{0}) + 1}{η_{1}^{∧}(1, π, δ_{0}) + 1}|^{δ_{0}} |\frac{η_{1}^{∧}(1, ψ, δ_{0}) − 1}{η_{1}^{∧}(1, π, δ_{0}) − 1}|^{δ_{0}} < 1.

Proof. According to the Lemma 4.21 in [42],

(5.14)  |\frac{η_{1}^{∧}(1, ψ, δ_{0})}{|η_{1}^{∧}(1, ψ^{∗}, δ_{0})}|^{1−δ_{0})} |\frac{η_{1}^{∧}(1, ψ, δ_{0}) + 1}{η_{1}^{∧}(1, ψ^{∗}, δ_{0}) + 1}|^{δ_{0}} |\frac{η_{1}^{∧}(1, ψ, δ_{0}) − 1}{η_{1}^{∧}(1, ψ^{∗}, δ_{0}) − 1}|^{δ_{0}} < 1.

Multyplying (5.1) and (5.14) we obtain (5.13). ■

Corollary 2. If δ_{0} ≤ 1/7 then

(5.15) |\frac{η_{1}^{∧}(1, π/3, δ_{0})}{|η_{1}^{∧}(1, π, δ_{0})}|^{1−δ_{0})} |\frac{η_{1}^{∧}(1, π/3, δ_{0}) + 1}{η_{1}^{∧}(1, π, δ_{0}) + 1}|^{δ_{0}} |\frac{η_{1}^{∧}(1, π/3, δ_{0}) − 1}{η_{1}^{∧}(1, π, δ_{0}) − 1}|^{δ_{0}} < 1.

Proof. Clearly, if ψ = π/3, then s = cos(ψ/2) = \frac{\sqrt{3}}{2} > 1/28 ≥ δ_{0}/4. Therefore, according to the Corollary 1, (5.15) holds. ■

Corollary 3. The following inequality holds:

(5.16)  |h^{∼}(η_{1}^{∧}(1, π, δ_{0})| > |h^{∼}(η_{1}^{∧}(1, π/3, δ_{0})|.

Proof. In view of (4.28),

(5.17)  |h^{∼}(η)| = 2^{−6}(1 − δ_{0})^{−3Δ+3} (|η^{2} − 1|^{δ_{0}}|η|^{1−δ_{0}})^{3/δ_{0}}

Therefore (5.16) follows from the Corrollary 2. ■

Lemma 6.5.2. Let

(5.18)  ρ_{1}(δ_{0}) = \frac{(2δ_{0})^{δ_{0}} (1 + \frac{3\sqrt{δ_{0}}}{2} )}{(1 + δ_{0})^{1+δ_{0}}(1 − δ_{0})^{1−δ_{0}}}

If 0 < δ_{0} ≤ 1/7, then

(5.19)  ρ_{1}(δ_{0}) > 1.

Proof . In view of (5.18),

ln(ρ_{1}(δ_{0}) = δ_{0} ln(δ_{0}) + δ_{0} ln(2) + ln (1 + \frac{3\sqrt{δ_{0}}}{2} ) −

(1 − δ_{0}) ln(1 − (δ_{0})^{2}) − 2δ_{0} ln(1 + δ_{0}) >
δ_{0}(− ln(1/δ_{0}) + \sqrt{1/δ_{0}}) + \sqrt{δ_{0}} − 9δ_{0}/8 − 2δ_{0}^{2} > 0,

if 0 < δ_{0} ≤ 1/7. ■

Corollary 1. If 0 < δ_{0} ≤ 1/7, then

(5.20)  |h^{∼}(η_{0}^{∧}(1, ψ^{∗}, δ_{0}))| > 1.

Proof. In view of (5.2), (5.8) and (5.9),

(5.21)  |η_{0}^{∧}(1, ψ^{∗}, δ_{0})|^{2(1−δ_{0})} × |(η_{0}^{∧}(1, ψ^{∗}, δ_{0}) − 1)(|η_{0}^{∧}(1, ψ^{∗}, δ_{0}) + 1)|^{2δ_{0}} ≥

(\frac{2δ_{0}}{1 + δ_{0}} )^{δ_{0}} (\frac{1 + \frac{δ_{0}}{2} + \frac{3\sqrt{δ_{0}}}{2}}{1 + δ_{0}} )^{1−δ_{0}} (\frac{4 (1 + \frac{δ_{0}}{4} + \frac{3\sqrt{δ_{0}}}{2} )}{1 + δ_{0}} )^{δ_{0}} >

4^{δ_{0}}(1 − δ_{0})^{1−δ_{0}}ρ_{1}(δ_{0}).

The inequality (5.20) follows from (5.21) and (5.17). ■

Corollary 2. If 0 < δ_{0} ≤ 1/7 and s = cos(ψ/2) ≥ δ_{0}/4, then

(5.22)  |h^{∼}(η_{0}^{∧}(1, ψ, δ_{0}))| > 1.

Proof. The inequality (5.22) follows from (5.20), (5.17) and Corollary to the Lemma 4.20 in [42]. ■

Corollary 3. If 0 < δ_{0} ≤ 1/7, then

(5.23)  |h^{∼}(η_{0}^{∧}(1, π/3, δ_{0}))| > 1.

Proof.Clearly, if ψ = π/3, then s = cos(ψ/2) = \frac{\sqrt{3}}{2} > 1/28 ≥ δ_{0}/4. Therefore, according to the Corollary 2, (5.23) holds. ■

According to (4.29), (5.16) and (5.23),

(5.24)  |h^{∼}(η_{0}^{∧}(1, π/3, δ_{0}))| = |h^{∼}(η_{0}^{∧}(1, π/3, δ_{0}))| > 1 >
|h^{∼}(η_{1}^{∧}(1, π, δ_{0})| = |h^{∼}(γ_{1})| >
|h^{∼}(η_{1}^{∧}(1, π/3, δ_{0}))| = |h^{∼}(η_{1}^{∧}(1, π/3, δ_{0}))|.

In view of (4.30), (5.24), and according to the Perron’s Theorem,

(5.25)  lim sup_{ν→∞} |f_{1}^{∗}(1; ν)|^{1/ν} = |h^{∼}(η_{0}^{∧}(1, π/3, δ_{0}))|.

According to the Lemma 4.2.1 in [39],

(5.26)  lim_{ν→∞} |f_{2}^{∗}(1; ν)|^{1/ν} = |h^{∼}(η_{0}^{∧}(1, π, δ_{0}))|,
(5.26)  lim_{ν→∞} |f_{2+2k}^{∗}(1; ν)/f_{2}^{∗}(1; ν)| = 0,

where k = 1, 2. Therefore

(5.27)  lim sup_{ν→∞} |f_{2+2k}^{∗}(1; ν)|^{1/ν} = |h^{∼}(η_{1}^{∧}(1, π/3, δ_{0}))|,

where k = 1, 2.

§6. Some corrections to the previous parts.

During of the work on this part I found some misprints in the previous parts. On the page 16, line 7 from boden in [35] must stand

R_{0}(t; ν)^{3} = ∑_{k=0}^{νΔ} α_{ν,k}^{∗}(t + k)^{−3} + ∑_{k=0}^{νΔ} β_{ν,k}^{∗}(t + k)^{−2} + ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}(t + k)^{−1}

nstead

R_{0}(t; ν)^{3} = ∑_{k=0}^{νΔ} α_{ν,k}^{∗}(t + k)^{−3} + ∑_{k=0}^{νΔ} β_{ν,k}^{∗}(t + k)^{−2} + ∑_{k=0}^{νΔ} γ_{ν,k}^{∗}(t + k)^{−3}

On the page 18, line 7 from the boden in [35] must stand

= 3α^{∗}(z; ν)L_{4}(z^{−1}) + 2β^{∗}(z; ν)L_{3}(z^{−1}) + γ^{∗}(z; ν)L_{2}(z^{−1}) − ψ^{∗}(z; ν),

instead

= 3α^{∗}(z; ν)L_{4}(z^{−1}) + 2β^{∗}(z; ν)L_{3}(z^{−1}) + γ^{∗}(z; ν)L_{2}(z)^{−1} − ψ^{∗}(z; ν),

On the page 9 line 7-8 in [39] from the top must stand

If z = 1, then condition of the Lemma 4.1.3 is fulfilled for J = {0, 1, 3, 4, 5}. instead If z = 1, then condition of the Lemma 4.1.3 is fulfilled for J = {0, 1, 2, 4, 5}. On the page 16, inequality (120) in [42] must stand

(120)  |η_{0}^{∧}(r, ψ, δ_{0})|^{1−δ_{0}}|(η_{0}^{∧}(r, ψ, δ_{0}))^{2} − 1|^{δ_{0}} >

instead

(120)  |η_{0}^{∧}(r, ψ, δ_{0})|^{1−δ_{0}}|(η_{0}^{∧}(r, ψ, δ_{0}))^{2} − 1|^{δ_{0}} ≥

On the page 24 in [36], equalities (2.4.7) must stand

(2.4.7)  D_{1}^{∼}(z, w) = (1/(z − 1))D^{∼}(z, w) = w^{6} − ∑_{k=1}^{6} b_{(}^{∼}z)w^{k−1}

insead

(2.4.7)  D_{1}^{∼}(z, w) = (1/(z − 1))D^{∼}(z, w) = (ν^{−1}w)^{6} − ∑_{k=1}^{6} b_{(}^{∼}z)w^{k−1}

On the page 24 in [36], equalities (2.4.8) must stand

(2.4.8)  det(B^{∼}(z) − d_{2}E) = D_{1}^{∼}(z, d_{2}) = (−1/(z − 1))(Δ(Δ + 1))^{3}.

insead

(2.4.8)  det(B^{∼}(z) − d_{2}E) = D_{1}^{∼}(z, d_{2}) = (1/(z − 1))(Δ(Δ + 1))^{3}.

On the page 27 in [36], equality (2.4.11) must stand

(2.4.14)  det(Q^{∼}(B^{∼}(z))) = ((−1/(z − 1))(Δ(Δ + 1))^{3})^{3d_{2}}d_{1}^{3d_{1}},

instead

(2.4.14)  det(Q^{∼}(B^{∼}(z))) = (−1/(z − 1))(Δ(Δ + 1))^{3})^{3d_{2}}d_{1}^{3d_{1}},

On the page 27, line 3 from bottom, equality (3.1.92) in [40] must stand

(3.1.92) D^{∼}(z, w) = −(η − 1)^{−m−n}(d^{∧})^{m+n}((η + δ)^{m+n} − (1 + δ)^{m+n}θ_{0}(z)η^{n})

instead

(3.1.92) D^{∼}(z, w) = −(η − 1)^{−m−n}(δ^{∧})m + n((η + δ_{1})^{m+n} − (1 + δ)^{m+n}θ_{0}(z)zη^{n}

References

[1] Apéry R., Interpolation des fractions continues et irrationalite de certaines constantes No 3 (1981), Bulletin de la section des sciences du C.T.H.S, 37 – 53.

[2] E.B. Van Vleck., On linear criteria for the determination of the radius of convergence of a power series 1:3 (1900), Transactions of the American Mathematical society, 293 – 309.

[3] O. Perron, Über die Poincareshe Differenzenglechung 137 (1910), Journal für die reine und angewandte Mathematik, 6 – 64.

[4] W. Maier, Potenzreihen irrationalen Grenzwertes, J.reine angew. Math. 156 (1927), 93 – 148.

[5] E.M. Nikis̆in, On irrationality of the values of the functions F(x,s) (Russian), Mat.Sb. 109 (1979), 410 – 417; English transl. in Math. USSR Sb. 37 (1980), 381 – 388..

[6] G.V. Chudnovsky, Pade approximations to the generalized hyper-geometric functions I, J.Math.Pures Appl. 58 (1979), 445 – 476..

[7] , Transcendental numbers 751 (1979), Number Theory,Carbondale,Lecture Notes in Math,,Springer-Verlag, 45 – 69..

[8] , Approximations rationelles des logarithmes de nombres rationelles, C.R.Acad.Sc. Paris,Série A t.288 (1979), 607 – 609.

[9] , Formules d’Hermite pour les approximants de Padé de logarithmes et de fonctions binômes, et mesures d’irrationalité, C.R.Acad.Sc. Paris,Série A t.288 (1979), 965 – 967.

[10] , Un systḿe explicite d’approximants de Padé pour les fonctions hypérgéometriques généraliesées,avec applications a l’arithmétique, C.R.Acad.Sc. Paris,Série A t.288 (1979), 1001 – 1004.

[11] , Recurrenses defining Rational Approximations to the irrational numbers, Proceedings of the Japan Academie,Ser. A, 58 (1982), 129 – 133.

[12] Alladi K. and Robinson M., Legendre polinomials and irrationality 318 (1980), J. Reine Angew.Math, 137 – 155..

[13] Dubitskas A,1986., An approximation of logarithms of some numbers (1986), Diophantine approximations II,Moscow, 20 – 34.

[14] ,1987., On approximation of π/\sqrt{3} by rational fractions, Vestnik MGU,series 1 No 6 (1987), 73 – 76.

[15] S.Eckmann, Über die lineare Unaqbhangigkeit der Werte gewisser Reihen, Results in Mathematics, 11 (1987), 7 – 43.

[16] Hata M.,1990., Legendre type polinomials and irrationality mesures, J. Reine Angew. Math. 407 (1990), 99 – 105.

[17] A.O. Gelfond., Transcendental and algebraic numbers. (Russian), GIT-TL, Moscow, 1952.

[18] Bateman H. and Erdélyi A., Higher transcendental functions, New-York – Toronto – London, Mc. Grow-Hill Book Company, Inc., 1953.

[19] N.E. Nörlund., Differenzenrechnung, Springer Verlag, 1924.

[20] Titchmarsh E. C., The Theory of Functions, Oxford University Press, 1939.

[21] Whittaker E.T. and Watson G.N., A course of modern analysis, Cambridge University Press, 1927.

[22] L.A.Gutnik L., On the Irrationality of Some Quantyties Containing ζ(3) (Russian), Uspekhi Mat. Nauk 34, 3(207) (1979), 190.

[23] , Eleven papers translated from the Russian, vol. 140, American Mathematical Society, 1988, pp. 45 - 56.

[24] , Linear independence over Q of dilogarithms at rational points (Russian), Uspekhi Mat.Nauk 37 (1982), 179-180; English transl. in Russ. Math. surveys 37 (1982), 176-177..

[25] , On the rank over ℚ of some real matrices 5736-84 (1984), VINITI, 1 – 29.

[26] , On the rank over ℚ of some real matrices 27 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 32.

[27] , On a measure of the irrationality of dilogarithms at rational points 4345-84 (1984), VINITI, 1 – 74.

[28] , To the question of the smallness of some linear forms 2413-B93 (1993), VINITI, 1 – 94.

[29] ,1995., About linear forms, whose coefficients are logarithms of algebraic numbers No 135-B95 (1995), VINITI, 1 – 149.

[30] , About systems of vectors, whose coordinates are linear combinations of logarithms of algebraic numbers with algebraic coefficients. 3122-B94 (1994), VINITI, 1 – 158.

[31] , About linear forms, whose coefficients are ℚ-proportional to the number log 2, and the values of ζ(s) for integer s, 3258-B96 (1996), VINITI, 1 – 70.

[32] , The lower estimate for some linear forms with coefficients ℕζ(1 + ℕ), 3072-B97 (1997), VINITI, 1 – 77.

[33] , About the behaviour of solutions of difference equations of Poincaré type 3384-B97 (1997), VINITI, 1 – 41.

[34] , On linear forms with coefficients in ℕζ(1 + ℕ), 3 (2000), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 13.

[35] , On linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 1) 15 (2001), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 20.

[36] , On linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 2) 104 (2001), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 36.

[37] , On some linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 3) 57 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 33.

[38] , On the dimension of some linear spaces over finite extension ℚ (part 2) 107 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 37.

[39] , On linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 4) 142 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 27.

[40] , On the dimension of some linear spaces over ℚ (part 3) 16 (2003), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 45.

[41] , On the difference equation of Poincaré type (Part 1). 52 (2003), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 43.

[42] , On the dimension of some linear spaces over ℚ (part 4) 73 (2003), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 37.

[43] , On linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 5) 83 (2003), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 13.

Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskij pereulok, 1-3/12, str.8 Moscow 109028 Russian Federation E-mail: gutnik@gutnik.mccme.ru