ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4).

L.A. Gutnik

Dedicated to the memory of Professor A.O. Gelfond.

Table of contents

§0. Foreword. §1. Some preparatory results. §2. Proof of the theorem 8. §3. Proof of the Theorem 9. §4. Proof of the Theorem 10. §5. The case of the general differrence equation of the Poincaré type.

§0. Foreword.

In [77]: was proved the following Theorem 7. Let s ∈ ℕ − 1, n ∈ ℕ,

a_{i}^{∼} ∈ ℂ, a_{i}(ν) ∈ ℂ,

a_{n}(ν) = 1, a_{i}(ν) − a_{i}^{∼} = O(1/(ν + 1)) (1)

for ν ∈ ℕ − 1 and i = 0, …, n. Let us consider the following difference equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (2)

where ν ∈ ℕ − 1.

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 2

For m ∈ ℕ let V_{m} denotes the linear over ℂ space of solutions y = y(ν) of the equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (3)

where ν ∈ m + ℕ − 1. Let the absolute values of all the roots of the characteristical polynomial

T (z) = ∑_{k=0}^{n} a_{k}^{∼}z^{k} (4)

are among the numbers {ρ_{i}: 1 ≤ i ≤ 1 + s} such that ρ_{s+1} = 0 and ρ_{j} < ρ_{i} for 1 ≤ i < j ≤ s + 1. Let e_{i} and k_{i} denote respectively the sum and the maximum of the multiplicities of those roots, whose absolute value is equal to the number ρ_{i}, where i = 1, …, s + 1, and let k^{∗} = k_{s+1}. We suppose that, if s > 0, then

e_{i} > 0 (5)

for i = 1 …, s. For given y = y(ν) in ℂ^{m−1+ℕ}, let

ω_{n,y}(ν) = max(|y(ν)| , …, |y(ν + n − 1)|).

Then there exist A > 0, m ∈ ℕ, α^{∧}(ν) > 0 with ν ∈ m + ℕ − 1 and the subspaces V_{m,1}^{∨}, …, V_{m,s+1}^{∨} such that

lim_{ν→∞} α^{∧}(ν) = 0,

V_{m} = V_{m,1}^{∨} ⊕ …, ⊕ V_{m,s+1}^{∨}, dim_{ℂ}(V_{m,i}^{∨}) = e_{i}, 1 ≤ i ≤ s + 1,

and, if y ∈ V_{m,θ}^{∨} for some θ ∈ {1, ..., s}, then

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν) (6)

for ν ∈ m + ℕ − 1; moreover, the spaces

V_{m,j}^{∧} = V_{m,j}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨},

where j = 1 …, s + 1, and, if s ≥ 1 natural projections π_{j},

V_{m,j}^{∧} ↦ V_{m,j}^{∨},

where j = 1 …, s, have the folloving properties:

if y ∈ V_{m,θ}^{∧} for some θ ∈ {1, ..., s}, then

ω_{n,y}(ν) ≤ exp(A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n,y}(m), (7)

(ω_{nπ_{θ(y)}}(m) − α(ν)ω_{n,y}(m))(ρ_{θ})^{ν} (8)

exp(−A(ln(ν) + ν^{1−1/k_{θ}})) ≤ ω_{n,y}(ν),

where ν ∈ m + ℕ − 1; if

k^{∗} > 0, (9)

and y ∈ V_{m,s+1}^{∨} (= V_{m,s+1}^{∧}), then

|y(ν)| ≤ (A/ν)^{ν/k^{∗}}ω_{n,y}(m), (10)

where ν ∈ m + ℕ − 1.

Remark 1. It follows from the Theorem 7 that the space V_{m,θ}^{∧}, where θ = 1, …, s + 1, does not depend from the construction and is defined uniquelly by means of the equality

V_{m,θ}^{∧} = {y ∈ V_{m}: lim sup |y(ν)|^{1/ν} ≤ ρ_{θ}}.

The presence of unkown α(ν) (even tending to zero) in (8) constrict the possibilities of the application of this Theorem. Of course, in view of (6), in the case e_{s+1} = 0, θ = s this α(ν) cannot play devil with us , because it vanishes then, but such happy case (see, for example, [75]-[76]) is rather an exeption from the rule. One may attempt to estimate the specified value but it would be better to get rid from it at all. With this goal I prove here the following

Theorem 10. Let are fulfilled all the conditions of the Theorem 7. Let further V be an arbitrary linear subspace of V_{m} such that

V ∩ V_{m,θ+1} = {0},

where θ ∈ {1 …, s}.

Then for this V there exists a constant A^{∗} = A^{∗}(V) > 0 such that

exp(−A^{∗}(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν) (11)

where y ∈ V, k = max(k_{1}, … k_{s}) and ν ∈ m + ℕ − 1.

First I prove the vollowing

Theorem 8. Let are fulfilled all the conditions of the Theorem 7. Let further

V_{m,j}^{∗} = V_{m,1}^{∨} ⊕ … ⊕ V_{m,j}^{∨},

where j = 1 …, s + 1, (and V_{m,1}^{∗} = V_{m,1}^{∨}.) Then vor V = V_{m,θ}^{∗} with θ ∈ {1, ..., s}, holds the assertion of the Theorem 10.

Then I prove

Theorem 9. Let for some θ ∈ {1 …, s} is given a linear map ξ_{θ} of the space V_{m,θ}^{∗} into V_{m,θ+1}^{∧}. Let I_{θ}^{∗} is the identity map V_{m,θ}^{∗} → V_{m,θ}^{∗} Then for

V = (I_{θ}^{∗} + ξ_{θ})(V_{m,θ}^{∗})

holds the assertion of the Theorem 10.

In the section 4 I prove the Theorem 10. And in the section 5 I discuss, what will take place, if instead (1) the

following conditions hold:

lim_{ν→∞} a_{i}(ν) = a_{i}^{∼}, (12)

where i = 0, …, n,

a_{n}(ν) = 1, (13)

where ν ∈ ℕ − 1.

Acknowledgments. I express my deepest thanks to Professors B.Z.Moroz, I.I. Piatetski-Shapiro, A.G.Aleksandrov, P.Bundshuh and S.G.Gindikin for help and support.

§1. Some preparatory results.

Lemma 1. Let a ∈ ℕ, b ∈ ℕ − 1 + a, C > 0 Then

0 < ∑_{κ=a}^{b} ln(1 + C/κ) ≤ (14)

ln(1 + C/a) + b ln(1 + C/b) − a ln(1 + C/a) + C ln((b + C)/(a + C)).

Proof. See the proof of the Lemma 1 in [77]. ■ Corollary. If a ∈ ℕ, b ∈ ℕ + a − 1, b < 2a, C > 0, then

∑_{κ=a}^{b} ln(1 + C/κ) ≤ 3C.

Proof. See the Proof of the Corollary of the Lemma 1 in [77]. ■ Lemma 2. ([51], Lemma 2, [44], Lemma 2, [72], Lemma 8.) Let A ∈ Mat_{n}(ℂ) an let k is a maximal order of its Jordan blocks. Then

there exists a constante γ^{∗}(A) > 0 with the following properties:

for any ε > 0 there exists a norm p_{A,ε} on ℂ^{n} such that

p_{A,ε} ≤ γ^{∗}(A)(max(1, 1/ε)^{k−1}h, (15)

h ≤ γ^{∗}(A)(max(1, ε)^{k−1}p_{A,ε}, (16)

(p_{A,ε})^{∼} ≤ (γ^{∗}(A))^{2}(max(ε, 1/ε)^{k−1}h^{∼}, (17)

h^{∼} ≤ (γ^{∗}(A))^{2}(max(ε, 1/ε)^{k−1}(p_{A,ε})^{∼}, (18)

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 5

∥A∥_{sp} ≤ (p_{A,ε})^{∼} ≤ ∥A∥_{sp} + (sign(k − 1))ε, (19) where ∥A∥_{sp} denotes the maximum of the absolute values of eigenvalues of the matrix A. If, moreover, det(A) ≠ 0, ∥A^{−1}∥_{sp}^{−1} > (sign(k − 1))ε, (20) then ∥A^{−1}∥_{sp} ≤ (21)

(p_{A,ε})^{∼}(A^{−1}) ≤ (∥A^{−1}∥_{sp}^{−1} − (sign(k − 1))ε)^{−1} =
(sign(k − 1)) ∥A^{−1}∥_{sp} + ε ∥A^{−1}∥_{sp} (∥A^{−1}∥_{sp}^{−1} − (sign(k − 1))ε)^{−1} .

Proof. See the proof of the Lemma 8 in [72]. ■

Corollary. If all the eigenvalues of the matrix A are simple, then (p_{A,ε})^{∼} = ∥A∥_{sp}. (22) If, moreover, det(A) ≠ 0, (23) then (p_{A,ε})^{∼}(A^{−1}) = (∥A^{−1}∥_{sp})^{−1} . (24)

Proof. See the proof of the Corollary of the Lemma 8 in [72]. ■

Lemma 3. Let all the conditions of the Theorem 7 are fulfilled, and let

k = max(k_{1}, …, k_{s}).

Then there exist A > 0, m ∈ ℕ such that ω_{n,y}(ν) ≤ exp(A(ln(ν) + ν^{1−1/k}))(ρ_{1})^{ν}ω_{n}(y)(m) (25) for any y ∈ V_{m} and ν ∈ m + ℕ − 1.

If, moreover, k^{∗} = 0, then there exist A > 0, m ∈ ℕ such that exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{s})^{ν}ω_{n,y}(m) ≤ ω_{n,y}(ν) (26) for any y ∈ V_{m} and ν ∈ m + ℕ − 1.

Proof. Since V_{m} = V_{m,1}^{∧}, it follows that the inequality (7) holds with θ = 1 for any y ∈ V_{m}. For the full proof of the Lemma let us make some not large changes in the proof of the Lemma 2 in [77].

The condition k^{∗} = 0 implies the inequality a_{0}^{∼} = T (0) ≠ 0, (27)

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 6

and for p in Theorem 6 of the paper [72] the equality p = n. Let

A(ν) = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −a_{0}(ν) & −a_{1}(ν) & −a_{2}(ν) & … & −a_{n−1}(ν) \end{matrix}) , (28)

where ν ∈ ℕ − 1, and let

A^{∼} = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −a_{0}^{∼} & −a_{1}^{∼} & −a_{2}^{∼} & … & −a_{n−1}^{∼}. \end{matrix}) (29)

Let λ_{j}, where j = 1, …, r, is the sequence of all the mutually distinct roots of the polynomial (4) and k_{j}^{∗} is the multiplicity of the root λ_{j}. Then

k = sup{k_{j}^{∗}: j = 1, …, r}.

Clearly,

∑_{j=0}^{r} k_{j}^{∗} = n,

ρ_{s} ≤ |λ_{j}| ≤ ρ_{1} = ∥A∥_{sp} < R = h^{∼}(A^{∼}) + 1,

where j = 1, …, r. In view of (1), there exists C_{1} > 0 such that h^{∼}(A(ν) − A^{∼}) ≤ C_{1}/(ν + 1). (30) Therefore, according to the Lemma 2, for any ε > 0 the following inequalities holds p_{A^{∼},ε}^{∼}(A(ν) − A^{∼}) ≤ (31)

(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}h^{∼}(A(ν) − A^{∼}) ≤
(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{1}/(ν + 1),

p_{A^{∼},ε}^{∼}(A(ν)) ≤ ρ_{1} + sign(k − 1)ε + (32)

(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{1}/(ν + 1) ≤

(ρ_{1} + (sign(k − 1))ε) (1 + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1} \frac{C_{1}}{ρ_{1}(ν + 1)}) ,

ln(p_{A^{∼},ε}^{∼}(A(ν))) ≤ ln(ρ_{1}) + ln(1 + (sign(k − 1))C_{1,1}ε) + (33)

ln (1 + (max(ε, 1/ε))^{k−1}C_{1,2}/(ν + 1)) ,

where

C_{1,1} = 1/ρ_{1}, C_{1,2} = (γ^{∗}(A^{∼}))^{2}C_{1}/ρ_{1}

and ν ∈ ℕ.

We consider first the case k > 1. For given ν ∈ ℕ + 1 we take d ∈ ℕ and a_{i} ∈ [1, ν] ∩ ℕ, where i = 0, …, d, in such a way that

a_{0} = 1, a_{d} = ν, a_{i−1} < a_{i} ≤ 2a_{i−1}, (34)

where i = 1, …, d, and

d ≤ \frac{ln(κ)}{ln(2)} + 1. (35)

According to the Corollary of the Lemma 1 and (34) – (35),

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}(A(κ))) ≤ (36)

(a_{i} − a_{i−1}) ln(ρ_{1}) + (a_{i} − a_{i−1}) ln(1 + C_{1,1}ε) +

∑_{κ=a_{i−1}}^{a_{i}−1} ln (1 + C_{1,2}(max(ε, 1/ε))^{k−1}/(κ + 1)) ≤

(a_{i} − a_{i−1})(ln(ρ_{1}) + ln(1 + ε/ρ_{1})) + 3C_{1,2}(max(ε, 1/ε))^{k−1},

where i = 1, …, d. We take now in (36) ε = ε_{i} = (a_{i−1})^{−1/k}. Then we obtain the inequality

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}(A(κ))) ≤

(a_{i} − a_{i−1}) ln(ρ_{1}) + (C_{1,1} + 3C_{1,2})(a_{i−1})^{1−1/k} =
(a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Therefore

ln(p_{A^{∼},ε_{i}}^{∼} ( ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)) ≤ (37)

(a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}

and, in view of (18),

ln(h^{∼} ( ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)) ≤ (38)

ln((γ^{∗}(A^{∼}))^{2})(max(ε, 1/ε))^{k−1}) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) ≤

2 ln(γ^{∗}(A^{∼})) + (i − 1)(1 − 1/k)ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) =

(i − 1)(1 − 1/k) ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Consequently,

ln (h^{∼} ( ∏_{κ=1}^{nu−1} A(ν − κ))) = (39)

ln (h^{∼} ((∏_{i=1}^{d} ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)))) =

= ν ln(ρ_{1}) + O(ν^{(1−1/k)}),

where ν ∈ ℕ.

If k = 1, then, according to (33),

ln(p_{A^{∼},1}^{∼}(A(ν))) ≤ ln(ρ_{1}) + C_{1,2}/(ν + 1),

ln (h^{∼} (∏_{κ=1}^{ν−1} A(ν − κ))) ≤ (40)

ln ((γ^{∗}(A^{∼}))^{2}p_{A^{∼},1}^{∼} (∏_{κ=1}^{ν−1} A(ν − κ))) ≤

ν ln(ρ_{1}) + O(ln(eν)).

where ν ∈ ℕ. In view of (39) – (40),

ln (h^{∼} (∏_{κ=1}^{ν−1} A(ν − κ))) = (41)

= ν ln(ρ_{1}) + O((ν + 1)^{(1−1/k)}) + O(ln(eν))),

where ν ∈ ℕ.

As in Section 3 of [72], let K denotes one of the fields ℝ or ℂ and L denotes a linear normed space over K with norm p = p(x). If L = K^{n}, we fix as p = p(x), wehre x ∈ K^{n}, the maximum of the absolute values of coordinates of the element x in the standard basis, i.e.

p(x) = h(x) = sup({|x_{1}|, …, |x_{n}|}), (42)

where

x = (x_{1} ; ⋮ ; x_{n}) .

If L is a Banach space with the norm p, then K−algebra of all the linear continuous operators acting in L will be denoted by 𝔐^{∧}(L), and the norm on 𝔐^{∧}(L), associated with the norm p will be denoted by p^{∼}. So,

p^{∼}(A) = sup({p(AX): X ∈ L, p(X) ≤ 1}).

It is well known that the associciated with h norm on Mat_{n}(ℂ) is defined as follows

h^{∼}(A) = sup ({∑_{k=1}^{n} |a_{i,j}|: i = 1, …, n}) , (43)

where A = (a_{i,k}) ∈ Mat_{n}(ℂ). The norms h and h^{∼} coincide respectiely with with the norms q_{∞} and q_{∞}^{∼} considered in section 6 of the paper [68].

Let m ∈ ℕ, and let E_{m}(L) be the set L^{m−1+ℕ} of all the maps of the set m − 1 + ℕ into L. The set E_{m}(L) is a linear space over K, where the muliplication of the elements by the number from K and addition of the elements are defined coordinate-wise. The subspace of E_{m}(L) composed by all the constant maps is isomorphic to the space L, and we identify this subspace with L.

As in Section 4 of [72], for any y ∈ E_{m}(ℂ) and n ∈ ℕ let Y_{n,y} and Y_{n,y}^{#} denote the elements in the space E_{m}(ℂ^{n}), wich are determined respectively by means the following equalities:

Y_{n,y}(ν) = ( y(ν) ; ⋮ ; y(ν + n − 1)) , (44)

Y_{n,y}^{#}(ν) = ( 0 ; ⋮ ; 0 ; y(ν)) , (45)

where ν ∈ m − 1 + ℕ. Let is fixed m ∈ ℕ. If y = y(ν) is a solution of the equation (3) for ν ∈ ℕ + m − 1, then

Y_{n,y}(ν) = ( ∏_{κ=1}^{ν−m} A(ν − κ)) Y_{n,y}(m),

where ν ∈ ℕ, and, in view of (41),

ω_{n,y}(ν) = h (Y_{n,y}(ν)) ≤ (46)

h^{∼} ( ∏_{κ=1}^{ν−m} A(ν − κ)) h (Y_{n,y}(m)) =

exp (O(1) (ν^{1−1/k} + ln(ν))) (ρ_{1})^{ν} ω_{n,y}(m),

where ν ∈ ℕ + m − 1. So, with m = 1 the asserted by the Lemma the upper estimate (25) of the value ω_{n,y}(ν) (ρ_{θ})^{−ν} = ω_{n,y}(ν) (ρ_{1})^{−ν} is obtained. We shall take now

ε ∈ (0, ρ_{s}/2). (47)

Then, in view of (21),

1/ρ_{s} ≤ (p_{A,ε})^{∼}(A^{−1}) ≤ (48)

1/ρ_{s} + 2(sign(k − 1))ε/ρ_{s}^{2} ≤ 2/ρ_{s}.

Let is fixed

m ∈ max([(2/ρ_{s})^{k}], [2C_{1}h^{∼} ((A^{∼})^{−1})]) + ℕ. (49)

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 10

Then h^{∼} ((A^{∼})^{−1} (A(ν) − A^{∼})) ≤ h^{∼} ((A^{∼})^{−1}) C_{1}/(ν + 1) ≤ (50)

h^{∼} ((A^{∼})^{−1}) C_{1}/([2C_{1}h^{∼} ((A^{∼})^{−1})] + 1) ≤ 1/2,

if ν ∈ m − 1 + ℕ, the matrix E + ((A^{∼})^{−1}(A(ν) − A^{∼})) is invertible,

h^{∼} ((E + ((A^{∼})^{−1}(A(ν) − A^{∼})))^{−1}) ≤ 2,

if ν ∈ m − 1 + ℕ, there exists the matix

(A(ν))^{−1} = (E + ((A^{∼})^{−1} (A(ν) − A^{∼})))^{−1} (A^{∼})^{−1} ,

if ν ∈ m − 1 + ℕ, moreover,

h^{∼} ((A(ν))^{−1}) ≤ h^{∼} ((E + ((A^{∼})^{−1}(A(ν) − A^{∼})))^{−1}) h^{∼} ((A^{∼})^{−1}) ≤

2h^{∼} ((A^{∼})^{−1})

and, finally,

h^{∼} ((A(ν))^{−1} − (A^{∼})^{−1}) =
h^{∼} ((A(ν))^{−1} (A^{∼} − A(ν)) (A^{∼})^{−1}) ≤
h^{∼} ((A(ν))^{−1}) h^{∼} ((A^{∼} − A(ν))) h^{∼} ((A^{∼})^{−1}) ≤ C_{2}/(ν + 1),

where

C_{2} = (h^{∼} ((A^{∼})^{−1}))^{2} C_{1}

and ν ∈ m − 1 + ℕ. Therefore p_{A^{∼},ε}^{∼}((A(ν))^{−1} − (A^{∼})^{−1}) ≤ (51)

(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}h^{∼}((A(ν))^{−1} − (A^{∼}))^{−1} ≤
(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{2}/(ν + 1),

where ν ∈ m − 1 + ℕ. In view of (51) and (21), p_{A^{∼},ε}^{∼}((A(ν))^{−1}) ≤ 1/ρ_{s} + (52)

2(sign(k − 1))ε/ρ_{s}^{2} ≤ 2/ρ_{s} + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{2}/(ν + 1) ≤

(1/ρ_{s} + 2(sign(k − 1))ε/ρ_{s}^{2}) (1 + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}ρ_{s}C_{2}ρ_{s}/(ν + 1)) , ln(p_{A^{∼},ε}^{∼}((A(ν))^{−1})) ≤ ln(1/ρ_{s}) + (53)

ln(1 + (sign(k − 1))C_{2,1}ε) + ln (1 + (max(ε, 1/ε))^{k−1}C_{2,2}/(ν + 1)) ,

where

C_{2,1} = 2/ρ_{s}, C_{2,2} = (γ^{∗}(A^{∼}))^{2}C_{2}ρ_{s}

and ν ∈ ℕ − 1 + m. We take ν ∈ m − 1 + ℕ.

We consider first the case k > 1 again now. For given ν ∈ ℕ + m we take d ∈ ℕ and a_{i} ∈ [m, ν] ∩ ℕ, where i = 0, …, d, in such a way that

a_{0} = m, a_{d} = ν, a_{i−1} < a_{i} ≤ 2a_{i−1} ≤ m2^{i}, (54)

where i = 1, …, d, and

d ≤ \frac{ln(ν)}{ln(2)} + 1. (55)

According to the Corollary of the Lemma 1 and (54) – (55),

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}((A(κ))^{−1})) ≤ (56)

(a_{i} − a_{i−1}) ln(1/ρ_{s}) + (a_{i} − a_{i−1}) ln(1 + C_{2,1}ε) +

∑_{κ=a_{i−1}}^{a_{i}−1} ln (1 + (max(ε, 1/ε))^{k−1}C_{2,2}/(κ + 1)) ≤

(a_{i} − a_{i−1})(ln(1/ρ_{s}) + ln(1 + C_{2,1}ε)) + 3(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{2,2},

where i = 1, …, d. We take now in (56) ε = ε_{i} = (a_{i−1})^{−1/k}. Then, in view of (49),

a_{i−1} ≥ m > (2/ρ_{s})^{k}, (a_{i−1})^{1/k} > 2/ρ_{s}, ε_{i} < ρ_{s}/2,

where i = 1, …, d, and therefore (47) and (56) hold. Consequently,

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}((A(κ))^{−1})) ≤ (57)

(a_{i} − a_{i−1}) ln(1/ρ_{s}) + (C_{2,1} + 3C_{2,2})(a_{i−1})^{1−1/k} =
(a_{i} − a_{i−1}) ln(1/ρ_{s}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Therefore

ln(p_{A^{∼},ε_{i}}^{∼} ( ∏_{κ=a_{i−1}^{a_{i}−1}}(A(κ))^{−1}) ≤ (58)

(a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}

and,in view of (18),

ln(h^{∼} ( ∏_{κ=1}^{a_{i}−a_{i−1}}(A(κ))^{−11}) ≤ (59)

ln((γ^{∗}(A^{∼}))^{2})(max(ε_{i}, 1/ε_{i}))^{k−1}) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) ≤

2 ln(γ^{∗}(A^{∼})) + (i − 1)(1 − 1/k)ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) =

(i − 1)(1 − 1/k)ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Consequently,

ln (h^{∼} ( ∏_{κ=m}^{ν−1}(A(κ))^{−1})) = (60)

ln (h^{∼} ((∏_{i=1}^{d} ∏_{κ=1}^{a_{i}−a_{i−1}}(A(κ))^{−1}))) =

= ν ln(1/ρ_{s}) + O(ν^{(1−1/k)}),

where ν ∈ ℕ + m.

If k = 1, then, according to (53),

ln(p_{A^{∼},1}^{∼}(A(ν))) ≤ ln(1ρ_{s}) + C_{2,2}/(ν + 1),

ln (h^{∼} ( ∏_{κ=m}^{ν−1}(A(κ))^{−1})) ≤ (61)

ln ((γ^{∗}(A^{∼}))^{2}p_{A^{∼},1}^{∼} ( ∏_{κ=m}^{ν−1}(A(κ))^{−1})) ≤

ν ln(1/ρ_{s}) + O(ln(eν)).

where ν ∈ ℕ. In view of (60) – (61),

ln (h^{∼} ( ∏_{κ=m}^{ν−1} A(ν − κ))) = (62)

= ν ln(1/ρ_{s}) + O(ν^{(1−1/k)}) + O(ln(eν))),

where ν ∈ ℕ + m. Since

Y_{n,y}(m) = ( ∏_{κ=1}^{ν−m} A(ν − κ))^{−1} Y_{n,y}(ν) =

( ∏_{κ=m}^{ν−1} (A(κ))^{−1}) Y_{n,y}(ν),

it follows that

ω_{n,y}(m) = h (Y_{n,y}(m)) ≤ (63)

h^{∼} ( ∏_{κ=m}^{ν−1} (A(κ))^{−1}) h (Y_{n,y}(ν)) =

exp (O(1) (ν^{1−1/k} + ln(ν))) (ρ_{1})^{−ν} ω_{n,y}(ν),

where ν ∈ ℕ+m. So, with m from (49) the asserted by the Lemma the lower estimate of the value ω_{n,y}(ν) (ρ_{s})^{−ν} is obtained. ■

Remark 1. In the case s = 1, k^{∗} = 0 the assertions of the Lemma and Theorem 7 coincide.

Lemma 4 (Perron’s decomposition Lemma, [22], Hilfsatz 3)). Let the characteristic polynomial (4) is represented as product

T (z) = T_{1}(z)T_{2}(z), (64)

where

T_{1}(z) = ∑_{α=0}^{p} b_{α}^{∼}z^{α}, T_{2}(z) = ∑_{β=0}^{q} u_{β}^{∼}z^{β}, (65)

with b_{p}^{∼} = u_{q}^{∼} = a_{n}^{∼} = 1 and absolute value of each root of T_{1}(z) is greater than the absolute value of each root of T_{2}(z).

Then there exist m ∈ ℕ, b_{α}(ν) ∈ ℂ, α = 0, …, p, ν ∈ ℕ + m − 1, and u_{β}(ν) ∈ ℂ, β = 0, …, q, ν ∈ ℕ + m − 1, such that

lim_{ν→∞} b_{α}(ν) = b_{α}^{∼}, α = 0, …, p, b_{p}(ν) = 1, b_{0}(ν) ≠ 0, (66)

lim_{ν→∞} u_{β}(ν) = u_{β}^{∼}, β = 0, …, q, u_{q}(ν) = 1, (67)

where ν ∈ ℕ + m − 1, and, moreover, the equation (3) is equivalent to the equation

∑_{α=0}^{p} b_{α}(ν)y(ν + α) = r(ν),

where ν ∈ ℕ − 1 + m and r(ν) satisfies to the equation

∑_{β=0}^{q} u_{β}(ν)r(ν + beta) = 0

with ν ∈ ℕ − 1 + m.

Lemma 5. Let all the conditions of the Perron’s decomposition Lemma are fulfilled and a_{k}(ν) − a_{k}^{∼} = O(1/(ν + 1)), k = 0, …, n, when ν → ∞.

Then for b_{α}(ν) with α = 0, …, p, and u_{β}(ν) with β = 0, …, q, from the assertion of the Perron’s decomposition Lemma the following conditions are fulfilled (cf. with (66) and (67)):

b_{α}(ν) − b_{α}^{∼} = O(1/ν), α = 0, …, p, b_{0}(ν) = 1, b_{q}(ν) ≠ 0,
u_{β}(ν) − u_{β}^{∼} = O(1/ν), β = 0, …, q, u_{0}(ν) = 1,

where ν ∈ ℕ − 1 + m.

Proof. The Lemma is direct corollary of the Theorem 5 in [68]. ■

§2. Proof of the theorem 8.

We use below the notations of the section 3 in [77]. Let K be one of the fields ℝ or ℂ. Let L be a linear normed space over K with a norm p(x). In the case L = K^{n} we fix as p(x), wehre x ∈ K^{n}, the maximum of the absolute values ofcoordinates of x in the standard basis, i.e.

p(x) = h(x) = sup({|x_{1}|, …, |x_{n}|}), (68) where

x = (x_{1} ; ⋮ ; x_{n}) .

If L is a Banach space with the norm p, then K−algebra of all the linear continuous operators acting in L will be denoted by 𝔐^{∧}(L), and the norm on 𝔐^{∧}(L), associated with the norm p will be denoted by p^{∼}. So,

p^{∼}(A) = sup({p(AX): X ∈ L, p(X) ≤ 1}).

It is well known that the associciated with h norm on Mat_{n}(ℂ) is defined as follows

h^{∼}(A) = sup ({∑_{k=1}^{n} |a_{i,j}|: i = 1, …, n}) , (69)

where A = (a_{i,k}) ∈ Mat_{n}(ℂ). The norms h and h^{∼} coincide respectiely with the norms q_{∞} and q_{∞}^{∼} considered in [68], section 6. Let m ∈ ℕ − 1, and we denote by E_{m}(L) the set L^{m−1+ℕ} of all the maps of the set m − 1 + ℕ into L. The set E_{m}(L) is a linear space over K, where the muliplication of the elements by the number from K and addition of the elements is defined coordinate-wise. The subspace of E_{m}(L) composed by all the constant maps is isomorphic to L, and we identify this subspace with L.

We denote by 𝔐^{∨}(L) the space of all the K−linear maps of the space L in L. If ϕ ∈ 𝔐^{∨}(L) and ψ ∈ 𝔐^{∨}(L), then ϕ ∘ ψ denotes the composition of operators ϕ and ψ, so that (ϕ∘ψ)f = ϕ((ψf)) for each f ∈ L. For x ∈ E_{m}(L) let

p_{m,∞}(x) = sup({p(x(ν)): ν ∈ m − 1 + ℕ}.

Let further E_{m,∞}(L) = {x ∈ E_{m}(L): p_{m,∞}(x) ≠ ∞},

E_{m,0}(L) = {x ∈ E_{m}(L): lim_{ν→∞} p(x(ν)) = 0},

E\overrightarrow{m}(L) = L + E_{m,0}(L).

Clearly, the space E\overrightarrow{m}(L) consists of all the y ∈ E_{m}(L), for which there exists

lim(y) = lim_{ν→∞}(y(ν)).

Let m ∈ ℕ − 1, μ ∈ m − 1 + ℕ and ler r_{m,μ} be the operator of restriction of the elements y ∈ E_{m}(L) on te set m − 1 + ℕ. Clearly, the map r_{m,μ} is an epimorphism of the space E_{m}(L) onto the space E_{μ}(L). If L is a K-algebra, then E_{m}(L) is a K-algebra, where the muliplication and addition of the elements is defined coordinate-wise; so, in this case r_{m,μ} is an epimorphism of K-algebra E_{m}(L) onto K-algebra E_{μ}(L). If L be an algebra with unity, let L^{∗} denotes the group of all its invertible elements. Then

(L^{∗})^{m−1+ℕ} ⊂ L^{m−1+ℕ};

we denote below (L^{∗})^{m−1+ℕ} by E_{m}(L^{∗}). Clearly,

E_{m}(L^{∗}) = (E_{m}(L))^{∗}.

Let L = ℂ^{n}, y ∈ E_{m}(L), and let y_{i}(ν) denotes the i-th coordinate of the element y(ν), where i = 1, …, n, ν ∈ m − 1 + ℕ; then the space (E_{m}(ℂ))^{n} contains an element ω(y), which has y_{i}(ν) as the value of its i-th coordinate at the point ν ∈ m − 1 + ℕ. So we obtain the natural isomorphism ω of the algebra E_{m}(ℂ^{n}) onto (E_{m}(ℂ))^{n}. This map ω induces an isomorphism of the algebra E_{m}(Mat_{n}(ℂ)) onto Mat_{n}(E_{m}(ℂ)). If L is a K − algebra, then each element a ∈ E_{m}(L) determines an acting on the space E_{m}(L) K-linear operator μ_{a} ∈ 𝔐^{∨}(E_{m}(L)), which turns any y ∈ E_{m}(L) into μ_{a}y = ay. On the space E_{m}(L) acts also K-linear operator ▽ ∈ 𝔐^{∨}(L), which turns any element y ∈ E_{m}(L) in the ▽y ∈ E_{m}(L) such that

(▽y)(ν) = y(ν + 1)

for any ν ∈ m − 1 + ℕ. Let us consider the subring 𝔄_{m}(L) of the ring 𝔐^{∨}(𝔏) generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E_{m}(L). Clearly,

μ_{a} ∘ ▽^{r} ∘ μ_{b} ∘ ▽^{s} = μ_{a▽^{r}b_{k}} ∘ ▽^{r+s}, (70)

where {r, s} ⊂ ℕ − 1, {a, b} ⊂ E_{m}(L). For each α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}} are uniquelly defined the number deg(α) and representation of α in the form

α = ∑_{k=0}^{deg(α)} μ_{a_{k}} ∘ ▽^{k}, (71)

where a_{k} ∈ E_{m}(L) for k = 0, …, deg(α) and a_{deg(α)} ≠ 0_{E_{m}(L)}. Clearly, (71) may be rewritten in the form

α = ∑_{k=0}^{∞} μ_{a_{k}} ∘ ▽^{k}, (72)

where a_{k} = 0_{E_{m}(L)} for k ∈ deg(α) + ℕ. It follows from (70) that 𝔄_{m}(L) is a graduated algebra, and if

β = ∑_{r=0}^{p} μ_{b_{r}} ∘ ▽^{r} ∈ 𝔄_{m}(L), (73)

γ = ∑_{s=0}^{q} μ_{c_{s}} ∘ ▽^{s} ∈ 𝔄_{m}(L), (74)

then

βγ = ∑_{k=0}^{p+q} ∑_{\begin{matrix}≤r≤p \\ 0≤s≤q \\ r+s=k \end{matrix}} μ_{b_{r}▽^{r}c_{s}} ∘ ▽^{r+s}; (75)

clearly, deg(βγ) = deg(β)+deg(γ), if b_{p}(ν)^{r}c_{q}(ν+p) is different from 0 at least for one ν ∈ m−1+ℕ. Let 𝔄\overrightarrow{m}(L) be the ring generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E\overrightarrow{m}(L). Since ▽a ∈ E\overrightarrow{m}(L), if a ∈ E\overrightarrow{m}(L), it follows, in view of (70), that 𝔄\overrightarrow{m}(L) is a graduated subalgebra 𝔄\overrightarrow{m}(L) of the algebra 𝔄_{m}(L), each α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}} admits a representation in the form (71) with a_{k} ∈ E\overrightarrow{m}(L) for k = 0, …, deg(α) and a_{deg(α)} ≠ 0_{E_{m}(L)}; to each such α corresponds the limit operator

(76) lim(α) = ∑_{k=0}^{deg(α)} μ_{lim(a_{k})} ∘ ▽^{k},

and polynomial

P (α, z) = ∑_{k=0}^{deg(α)} lim(a_{k})z^{k} ∈ L[z]. (77)

If α = 0_{𝔄_{m}(L)}, then we put

lim(α) = 0_{𝔄_{m}(L)}, P (α, z) = 0_{L[z]}.

The equality (70) shows that the map

α → P (α, z) (78)

is an epimorphism of the algebra 𝔄\overrightarrow{m}(L) on the algebra L[z] (if the algebra L is noncommutative, then we can treat the algebra L[z] as a semigroup ring of the semigroup (ℕ − 1, +) over the algebra L). We note that, if α ∈ 𝔄_{m}(ℂ), then, clearly, Ker(α) coincides with the linear space of all the solutions of the equation (3), and, moreover, if α ∈ 𝔄\overrightarrow{m}(ℂ), then the corresponding to α equation (3) is an equation of the Poincar’e type and P (α, z) is its characterictical polynomial.

Let L be an algebra with unity. The set of all α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (71) with a_{deg(α)} ∈ E_{m}(L^{∗}) will be denoted further by 𝔄_{m}(L)^{∘}. The set of all the the elements α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (71) with a_{deg(α)} = 1_{E_{m}(L)} will be denoted further by 𝔄_{m}(L)^{∨}. The set of all the α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (71) with a_{0} ∈ E_{m}(L^{∗}), will be denoted further by 𝔄_{m}(L)^{∧}. Let further

𝔄_{m}(L)^{∘∧} = 𝔄_{m}(L)^{∘} ⋂ 𝔄_{m}(L)^{∧}, 𝔄_{m}(L)^{∨∧} = 𝔄_{m}(L)^{∨} ⋂ 𝔄_{m}(L)^{∧},

𝔄_{m}(L)^{∘→} = 𝔄_{m}(L)^{∘} ⋂ 𝔄_{m}(L)^{→}, 𝔄_{m}(L)^{∨→} = 𝔄_{m}(L)^{∨} ⋂ 𝔄_{m}(L)^{→},

𝔄_{m}(L)^{∘∧→} = 𝔄_{m}(L)^{∘∧} ⋂ 𝔄_{m}(L)^{→}, 𝔄_{m}(L)^{∨∧→} = 𝔄_{m}(L)^{∨∧} ⋂ 𝔄_{m}(L)^{→},

Clearly, 𝔄_{m}(L)^{∘} consists of epimorphisms of the space E_{m}(L) onto E_{m}(L). The above map r_{m,μ} induces epimorphism r_{m,μ}^{▷} of the algebra 𝔄_{m}(L) on the algebra 𝔄_{μ}(L) defined as follows:

if α ∈ 𝔄_{m}(L),

α = ∑_{k=0}^{n} μ_{a_{k}} ∘ ▽^{k}, (79)

then

r_{m,μ}^{▷}(α) = ∑_{k=0}^{n} μ_{r_{m,μ}(a_{k})} ∘ ▽^{k}, (80)

where the operator ▽ in (79) acts in E_{m}(L) and the operator ▽ in (80) acts in E_{μ}(L). Clearly, r_{m,μ}^{▷} surjectively maps

𝔄_{m}(L)^{∘} onto 𝔄_{μ}(L)^{∘}, 𝔄_{m}(L)^{∨} onto 𝔄_{μ}(L)^{∨},

𝔄_{m}(L)^{∧} onto 𝔄_{m}u(L)^{∧}, 𝔄_{m}(L)^{∘∧} onto 𝔄_{μ}(L)^{∘∧},

𝔄_{m}(L)^{∨∧} onto 𝔄_{μ}(L)^{∨∧}, 𝔄_{m}(L)^{∘→} onto 𝔄_{μ}(L)^{∘→},

𝔄_{m}(L)^{∨→} onto 𝔄_{μ}(L)^{∨→}, 𝔄_{m}(L)^{∘∧→} onto 𝔄_{μ}(L)^{∘∧→},

𝔄_{m}(L)^{∨∧→} onto 𝔄_{μ}(L)^{∨∧→}. Since the diagram

E_{m}(L) ⟶^{r_{m,μ}} E_{μ}(L)
α↓  ↓r_{m,μ}^{▷}(α)
E_{m}(L) ⟶_{r_{m,μ}} E_{μ}(L)

is commuative and therefore

r_{m,μ}α = r_{m,μ}^{▷}(α)r_{m,μ}, (81)

it follows that r_{m,μ} surjectively maps Ker(r_{m,μ}α) onto

Ker(r_{m,μ}^{▷}(α)) ⊃ r_{m,μ}Ker(α).

Lemma 6. If μ ∈ m − 1 + ℕ and α ∈ 𝔄_{m}(L)^{∧}, then the operator α bijectively maps Ker(r_{m,μ}) onto Ker(r_{m,μ}).

Proof. Proof is given in [72], Lemma 3.

Corollary 1. Let μ ∈ m − 1 + ℕ and let α ∈ 𝔄_{m}(L)^{∧}. If

g ∈ E_{m}(L), x ∈ E_{μ}(L), m ≤ μ, α ∈ 𝔄_{m}(L)^{∧},
r_{m,μ}(g) = (r_{m,μ}^{▷}(α))(x),

then there exists a unique y ∈ E_{m}(L) such that

α(y) = g, r_{m,μ}(y) = x;

Proof. Proof is given in [72], Corollary 1 to the Lemma 3.

Corollary 2. Let μ ∈ m − 1 + ℕ and α ∈ 𝔄_{m}(L)^{∧}. Then and r_{m,μ} bijectively maps Ker(α) onto Ker(r_{m,μ}^{▷}(α)) = r_{m,μ}(Ker(α)).

Proof. Proof is given in [72], Corollary 2 to the Lemma 3.

If for the equation (2) are fulfilled the conditions (1) then

a_{k} = (a_{k}(0), a_{k}(1), …, a_{k}(ν), …) ∈ E\overrightarrow{0}(ℂ), (82)

where k = 0, …, n. Moreover, a_{n} = 1_{E_{0}(ℂ)}, for α in (79) Ker(α) coincides with the linear over ℂ space of all the solutions of the equation (2), polynomial (4) is equal to the polynomial P (α, z) = P (r_{0,m}^{▷}(α), z), where m ∈ ℕ, and the set Ker(r_{0,m}^{▷}(α)) coincides with linear over ℂ space V_{m} of all the solutions of the equation (3).

Let 𝔳 be the element in E_{0,0}, for which

𝔳(ν) = \frac{1}{ν + 1},

where ν ∈ ℕ − 1. Clearly, r_{0,m}(𝔳)E_{m,∞}(ℂ) ⊂ E_{m,0}(ℂ) for any m ∈ ℕ − 1. Let

E_{m,0}^{≻}(L) = r_{0,m}(𝔳)E_{m,∞}(L), E_{m}^{≻}(L) = L + E_{m,0}^{≻}(L).

Let us consider the ring 𝔄_{m}^{≻}(L) generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E_{m}(L)^{≻}. The Lemma 5 may be reformulated as follows:

Lemma 7. Let α ∈ 𝔄_{0}^{≻}(ℂ) ⋂ 𝔄_{0}^{∨}(ℂ), and P (α, z) coincides with the polynomial T (z) in (4) and (64).

Then there exist m ∈ ℕ and representation of the operator r_{0,m}^{▷}(α) in the form

r_{0,m}^{▷}(α) = ηβ (83)

such that

η ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨}(ℂ), deg(η) = q (84)

β ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨∧}(ℂ), deg(β) = p = n − q, (85)

and the polynomials P (β, z) , P (η, z) coincide respectively with the polynomials T_{1}(z) , T_{2}(z) in (65) .

Lemma 8. Let are fulfilled all the conditions of the Theorem 7. Let α ∈ 𝔄_{0}^{≻}(ℂ) ⋂ 𝔄_{0}^{∨}(ℂ) corresponds to the equation (2), i.e. (with m = 0) (79) holds

α = ∑_{k=0}^{n} μ_{a_{k}} ∘ ▽^{k}, (86)

where a_{n} = 1_{E_{0}(ℂ)} and

a_{k} = (a_{k}(0), a_{k}(1), …, a_{k}(ν), …) ∈ E_{0}^{≻}(ℂ), (87)

for k = 0, …, n.

Let the characteristic polynomial (4) is represented as product

P (α, z) = T (z) = ∏_{k=1}^{s+1} T_{i}(z), (88)

where

T_{i}(z) = ∑_{α=0}^{e_{i}} b_{i,α}^{∼}z^{α}, (89)

with b_{i,e_{i}} = a_{n}^{∼} = 1 and absolute value of each root of the polynomial T_{i}(z) is equal to ρ_{i}.

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 19

Then there exist m ∈ ℕ and representation of the operator r_{0,m}^{▷}(α) in the form

r_{0,m}^{▷}(α) = ∏_{i=0}^{s} β_{s+1−i} (90)

such that β_{i} ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨}(ℂ), deg(β_{i}) = e_{i}, (91)

P (β_{i}, z) = T_{i}(z),

where i = 1, …, s + 1 and β_{i} ∈ 𝔄_{m}^{∧}(ℂ), (92) where i = 1, …, s.

Proof. The assertion of the Lemma may be obtained by means of the sequentielly applying of the Lemma 7. ■

Let

β_{θ}^{∗} = ∏_{i=0}^{θ−1} β_{θ−i}, β_{θ}^{∧} = ∏_{i=s+1−θ}^{s} β_{s+1−i}, (93)

where θ = 1, … s + 1. In view of (refeq:2cdj), r_{0,m}^{▷}(α) = β_{θ+1}^{∧}β_{θ}^{∗}, (94) θ = 1, … s.

Let C > 0, n ∈ ℕ,

w_{C,n}(ν) = (\frac{C}{ν} )^{ν/n} , (95)

where ν ∈ ℕ; let further ρ > 0 and v_{C,n,ρ}(ν) = ρ^{ν} exp (C ((ν)^{1−1/n} + ln(ν))) , (96) where ν ∈ ℕ.

Let θ_{1} ∈ {1, … s}.Replacing m by some bigger m and applying to β_{θ_{1}}^{∧} in (93) the Theorem 7, we see that there exist A^{∨} > 0, m ∈ ℕ, α^{∨}(ν) > 0 with ν ∈ m + ℕ − 1 and the subspaces R_{m,θ_{1},θ_{1}}^{∨}, …, R_{m,θ_{1},s+1}^{∨} of the space R_{m} = Ker(β_{θ_{1}}^{∧}) such that

lim_{ν→∞} α^{∨}(ν) = 0,

R_{m} = R_{m,θ_{1},θ_{1}}^{∨} ⊕ …, ⊕ R_{m,θ_{1},s+1}^{∨}, dim_{ℂ}(R_{m,θ_{1},i}^{∨}) = e_{i}, θ_{1}, ≤ i ≤ s + 1, if r ∈ R_{m,θ_{1},θ}^{∨} (97) for some θ ∈ {θ_{1}, , ..., s}, then exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))ω_{q,r}(m)(ρ_{θ})^{ν} ≤ ω_{q,r)}(ν), (98) where ν ∈ ℕ + m − 1; moreover, the spaces

R_{m,θ_{1},j}^{∧} = R_{m,θ_{1},j}^{∨} ⊕ … ⊕ R_{m,θ_{1},s+1}^{∨},

where j = θ_{1}, , …, s + 1 and natural projections π_{θ_{1},j}^{∨} of the space R_{m,θ_{1},j}^{∧} onto the space R_{m,θ_{1},j}^{∨}, where j = θ_{1}, , …, s, have the folloving properties:

if

r ∈ R_{m,θ_{1},θ}^{∧} (99)

for some θ ∈ {θ_{1}, , ..., s}, then

ω_{q,r}(ν) ≤ (ρ_{θ})^{ν} exp(A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))ω_{q,r}(m), (100)

ω_{q,r}(ν) ≥ (ω_{q,π_{θ_{1},θ}^{∨}(r)}(m) − α^{∨}(ν)ω_{q,r)}(m)) × (101)

(ρ_{θ})^{ν} exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}})),

where ν ∈ ℕ + m − 1; if e_{s+1} > 0, and

r ∈ R_{m,θ_{1},s+1}^{∨} (= R_{m,θ_{1},s+1}^{∧}), (102)

then

|r(ν)| ≤ (A^{∨}/ν)^{ν/k^{∗}}ω_{q,r}(m), (103)

where ν ∈ ℕ + m − 1.

Let

β = β_{1}^{∗} = β_{1}, η = β_{2}^{∧}. (104)

Then, in view of (94), (83) holds.

Lemma 9. Let all the conditions of the Theorem 7 are fulfilled. For sufficient big m ∈ ℕ there exists the splitting monomorphism

ψ_{2}^{∧}: Ker(η) → Ker(ηβ)

with following properties:

(a)

Ker(β) = V_{m,1}^{∨}, ψ_{2}^{∧}(Ker(η)) = V_{m,2}^{∧},

where V_{m,1}^{∨} and V_{m,2}^{∧} are defined in the assertion of the Theorem 7;

(b) the monomorphism ψ_{2}^{∧} maps isomorphically the space R_{m,2,j}^{∨}, where j = 2, , …, s + 1, onto the space V_{m,j}^{∨};

(c) the map β_{1}ψ^{w}edge_{2} coincides with identiti map Ker(η) → Ker(η);

(d) the restriction of the map ψ^{w}edge_{2}β_{1} on the space V_{m,2}^{∧} coincides with the identity map V_{m,2}^{∧} → V_{m,2}^{∧};

(e) let I be the identiti map V_{m} → V_{m}; then the natural projection π_{1} of the space V_{m,1}^{∧} = V_{m} onto V_{m,1}^{∨} coincides with restriction of the map I − ψ_{2}^{∧}β_{1} on the space V_{m,1}^{∨};

(f) if j ∈ [2, s] ∩ ℕ, then natural projection π_{j} of the space V_{m,j}^{∧} onto the spaceV_{m,j}^{∨} coincides with the restriction of the map ψ_{2}^{∧}π_{2,j}^{∨}β_{1} on the space V_{m,j}^{∧} and the projection π_{2,j}^{∨} coincides with the restriction of the map β_{1}π_{j}, ψ_{2}^{∧} on the space R_{2,j}^{∧}.

Proof. See the proof of the Theorem 7 in [77], especially the section 2 and section 3. ■.

Remark 2. I use this opportunity to make a corrections in [77]. On the page 13, third line from the bottom must stand ψ_{m}π_{j}^{∨}ϕ_{m} instead ψ_{m}π_{j}^{∨}; on the second line from the bottom must stand π_{j}^{∨} = ϕ_{m}π_{j}ψ_{m} instead π_{j}^{∨} = ϕ_{m}π_{j}.

Lemma 10. Let all the conditions of the Lemma 9 are fulfilled. Then (for sufficient big m)

V_{m,θ}^{∗} = Ker(β_{1,θ}^{∗}), (105)

where θ = 1, …, s.

Proof. We use induction on s For s = 1 or θ = 1 the assertion of the Lemma directly follows from the Lemma 9.

Let s > 1 and assertion of the lemma is true for s − 1. Let θ > 1. Since the monomorphism ψ_{2}^{∧} maps isomorphically the space R_{m,2,j}^{∨}, where j = 2, , …, θ, onto the space V_{m,j}^{∨} and the map β_{1}ψ_{2}^{∧} coincides with identity map Ker(η) → Ker(η), it follows that the restriction of the map β_{1} on the space V_{m,2}^{∨} ⊕ … ⊕ V_{m,θ}^{∨} is an isomorphism of V_{m,2}^{∨} ⊕ … ⊕ V_{m,θ}^{∨} onto R_{m,2,θ}^{∗}. According to the inductive hypothesis,

R_{m,2,θ}^{∗} = Ker(β_{2,θ}^{∗}).

Therefore, if y ∈ V_{m,2}^{∨} ⊕ … ⊕ V_{m,θ}^{∨} then β_{1,θ}^{∗}(y) = β_{2,θ}^{∗}(β_{1}(y)) = 0; moreover, since also V_{m,1}^{∨} = Ker(β_{1}), it follows that V_{m,θ}^{∗} ⊂ Ker(β_{1,θ}^{∗}). Since, according to the Theorem 7,

dim_{ℂ}(V_{m,θ}) = ∑_{i=1}^{θ} e_{i} = (106)

dim_{ℂ}(Ker(β_{1,θ}^{∗})) = n_{θ} := deg(β_{1,θ}^{∗}),

it follows that (105) holds. ■.

Proof of the Theorem 8. In view of (105), (106), we apply to β_{1,θ}^{∗} the Lemma 3. Since θ plays the role of s in the lemma 3 now and

max(k_{1}, …, k_{θ}) ≤ k = max(k_{1}, …, k_{s}),

it follows that there exist A = A_{θ} > 0, m ∈ ℕ such that

exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{s})^{ν}ω_{n_{θ},y}(m) ≤ (107)

ω_{n_{θ},y}(ν) ≤ ω_{n,y}(ν)

for any y ∈ V_{m,θ}^{∗}. Since both the functions y → ω_{n_{t}heta,y}(m), y ∈ V_{m,θ}^{∗}, and y → ω_{n,y}(m), y ∈ V_{m,θ}^{∗} are two norms on the n_{θ}-dimensional space V_{m,θ}^{∗} there exists B = B_{θ} > 0 such that

ω_{n,y}(m) exp(−B) ≤ ω_{n_{θ},y}(m)

for any y ∈ V_{m,θ}^{∗}. Then

exp(−A(ln(ν) + ν^{1−1/k}) − B)(ρ_{θ})^{ν}ω_{n,y}(m) ≤ ω_{n,y}(ν) (108)

for any y ∈ V_{m,θ}^{∗}. ■

L.A.Gutnik, ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 4). 22

§3. Proof of the theorem 9.

Theorem 9. Let for some θ ∈ {1 …, s} is given a linear map ξ_{θ} of the space V_{m,θ}^{∗} into V_{m,θ+1}^{∧}. Let I_{θ}^{∗} is the identity map V_{m,θ}^{∗} → V_{m,θ}^{∗} Then for

V = (I_{θ}^{∗} + ξ_{θ})(V_{m,θ}^{∗})

holds the assertion of the Theorem 10.

Proof of the theorem 9. Since V_{m,θ}^{∗} is finite-dimensional linear space over ℂ, it follows that the map ξ_{θ} is continuous. Therevore there exists C > 0 such that ω_{n_{θ},ξ_{θ}(y)}(m) ≤ exp(C)ω_{n_{θ},ξ_{θ}(y)}(m) (109) for any y ∈ V_{m,θ}^{∗}. According to the Theorem 7 and Theorem 8, there exist a numbers A > 0, m ∈ ℕ, such that, if y ∈ V_{m,θ}^{∗}, ν ∈ m − 1 + ℕ, then ω_{n,ξ_{θ}(y)}(ν) ≤ exp(A(ln(ν) + ν^{1−1/k_{θ+1}}))(ρ_{θ})^{ν}ω_{n,ξ_{θ}(y)}(m) (110) for θ < s, ω_{n_{θ},ξ_{θ}(y)}(m) = 0 (111) for θ = s and k^{∗} = k_{s+1} = 0, |ξ_{θ}(y)(ν)| ≤ (A/ν)^{ν/k^{∗}}ω_{n,ξ_{θ}(y)}(m) (112) for θ = s and k^{∗} = k_{s+1} > 0, exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m) ≤ ω_{n,y}(ν). (113) Therefore

ω_{n,y+ξ_{θ}(y)}(ν) ≥ ω_{n,y}(ν) − ω_{n,ξ_{θ}(y)}(ν) ≥ exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m) (114)

for θ = s and k^{∗} = k_{s+1} = 0, ω_{n,y+ξ_{θ}(y)}(ν) ≥ ω_{n,y}(ν) − ω_{n,ξ_{θ}(y)}(ν) ≥ (115)

exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)−
|ξ_{θ}(y)(ν)| ≤ (A/ν)^{ν/k^{∗}} exp(C)ω_{n,y}(m) =
exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)×
(1 + exp(A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{−ν}(A/ν)^{ν/k^{∗}} exp(C))

for θ = s and k^{∗} = k_{s+1} > 0, ω_{n,y+ξ_{θ}(y)}(ν) ≥ ω_{n,y}(ν) − ω_{n,ξ_{θ}(y)}(ν) ≥ (116)

exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)−
(A/ν)^{ν/k^{∗}} exp(C)ω_{n,y}(m) =
exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)×

(1 − exp(A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{−ν}(A/ν)^{ν/k^{∗}} exp(C))

for θ = s and k^{∗} = k_{s+1} > 0,

ω_{n,y+ξ_{θ}(y)}(ν) ≥ ω_{n,y}(ν) − ω_{n,ξ_{θ}(y)}(ν) ≥ (117)

exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)−
exp(A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν} exp(C)ω_{n,y}(m) =
exp(−A(ln(ν) + ν^{1−1/k}))(ρ_{θ})^{ν}ω_{n,y}(m)×
(1 − exp(2A(ln(ν) + ν^{1−1/k}))(ρ_{θ+1}/ρ_{θ})^{ν} exp(C))

for θ < s.■

Remark 3. The value of ω_{n,y}(m) in (8) may be much bigger than the value of ω_{nπ_{θ(y)}}(m). Therefore we cannot (as show a simple examples) for fixed value of A in the inequality (11) to replace the linear space V of the Theorem 10 by the set V_{m}⧹V_{m,θ+1}^{∧}. But in (116) – (117) ω_{n,y}(m) is carried out the brackets, and the value in the brackets tends to 1, when ν → +∞, being greather than 1/2 for ν ∈ m − 1 + ℕ, with sufficient big m ∈ ℕ, which depends only from the equation.

§4. Proof of the theorem 10.

Proof of the theorem 10. Let π^{∗} and π^{∧} are restrictions on V of the natural endomorphisms of the space V_{m} onto respectively V_{m,θ}^{∗} and V_{m,θ+1}^{∧} and let I_{0} be the identity map V → V. Then

I_{0} = π^{∗} + π^{∧} (118)

Since Ker(π^{∗}) ⊂ V ∩ V_{m,θ+1}^{∧} = {0}, it follows that π^{∗} is an isomorphism of the space V onto linear subspace V′ of the space V_{m,θ}^{∗}. Let τ be the inverse isomorphism to the isomorphism π^{∗}. Then π^{∗}τ is identity map V′ → V′ and it follows from (118) that

τ = π^{∗}τ + π^{∧}τ. (119)

Clearly, the linear map π^{∧}τ : V′ → V_{m,θ+1}^{∧} have an extension

ξ_{θ} : V_{m,θ}^{∗} → V_{m,θ+1}^{∧}. (120)

It follows from (119) that

V = τ(V′) ⊂ (I + ξ_{θ})V_{m,θ}^{∗},

where I is identity map V_{m,θ} → V_{m,θ} and ξ_{θ} is a linear map in (120).

So, the Theorem 10 is Corollary of the Theorem 9. ■

§5. The case of the general differrence equation of the Poincaré type.

If (12) and (13) hold instead of (1), then making use of the above arguments we obtain the following chainges in the Theorem 7 and theorem 10. For any ε > 0 there exists a constantA^{∧} = A^{∧}(ε) > 0 such that, if y ∈ V_{m,θ}^{∧} with some θ ∈ {1, ..., s}, then (instead of (7) the following inequality holds)

ω_{n,y}(ν) ≤ exp(A^{∧}(ρ_{θ} exp(ε))^{ν}ω_{n,y}(m), (121)

where ν ∈ m + ℕ − 1; if y ∈ V_{m,s+1}^{∨} (= V_{m,s+1}^{∧}), then (instead of (10) the following inequality holds)

ω_{n,y}(ν) ≤ exp(A^{∧}(exp(−ε))^{ν}ω_{n,y}(m), (122)

where ν ∈ m + ℕ − 1.

Let further V be an arbitrary linear subspace of V_{m} such that

V ∩ V_{m,θ+1} = {0},

where θ ∈ {1 …, s}. Then for this subspace V and any ε > 0 there exists a constant A^{∨} = A^{∨}(V, ε) > 0 such that (instead of (11) the following inequality holds)

exp(−A^{∨})(ρ_{θ} exp(−ε))^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν) (123)

where y ∈ V and ν ∈ m + ℕ − 1.

Corollary. (See [51], Theorem 3 and [75], Lemma 16). Let as before (12) and (13) hold instead of (1). Let V be a r-dimensional subspace of V_{m}, let

V ∩ V_{m,s+1}^{∨} = {0}

and let {y_{1}(ν) , …, y_{r}(ν)} be a basis of the space V. Let

k_{3}(V) = max{k ∈ ℤ: 1 ≤ k ≤ s, V ⊂ V_{m,k}^{∧}},

and

k_{4}(V) = min{k ∈ ℤ: 1 ≤ k ≤ s, V ∩ V_{m,k+1}^{∧} = {0}}.

For X = (x_{1} , …, x_{r}), X ∈ ℂ^{r}, let

h(X) = max{|x_{1}| , …, |x_{r}|},
y = y^{∨}(X, ν) = x_{1}y_{1}^{∨}(ν) + … + x_{r}y_{r}^{∨}(ν).

Then for every ε ∈ (0, 1) there exist C_{3}(ε) > 0 and C_{4}(ε) > 0 such that

C_{4}(ε)(ρ_{k_{4}}(1 − ε))^{ν}h(X) ≤ ω_{n,y}(ν) ≤ C_{3}(ε)(ρ_{k_{3}} + ε)^{ν}h(X).

Proof The functions h(X) and the restriction of ω_{n,y}(m) on V are two norms on the r-dimensional over ℂ linear space V. Therefore there exists a constant C_{5} > 0 such that h(X) ≤ C_{5}ω_{n,y}(m) and ω_{n,y}(m) ≤ C_{5}h(X) The assertion of the Corollary directly follows from (121) and (123) now. ■

References.

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

[2] F.Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., 1979, 11, 268 – 272;

[3] A.van der Porten, A proof that Euler missed...Apéry’s proof of the irrationality of ζ(3), Math Intellegencer, 1979, 1, 195 – 203;

[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) (in 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, Number Theory,Carbondale, Lecture Notes in Math, Springer-Verlag, 1979, 751, 45 – 69;

[8] , Approximations rationelles des logarithmes de nombres rationelles C.R.Acad.Sc. Paris, Série A, 1979, 288, 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, 1979, t.288, 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, 1979, t.288, 1001 – 1004;

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

[12] , On the method of Thue-Siegel, Annals of Mathematics, 117 (1983), 325 – 382;

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

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

[15] , On approximation of π/\sqrt{3} by rational fractions, Vestnik MGU, series 1, 1987, 6, 73 – 76;

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

[17] M.Hata, Legendre type polinomials and irrationality mesures, J. Reine Angew. Math., 1990, 407, 99 – 125;

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

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

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

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

[22] O.Perron, Über die Poincaresche Differenzengleichumg, Journal für die reine und angewandte mathematik, 1910, 137, 6 – 64;

[23] A.O.Gelfond, Differenzenrechnung (in Russian), 1967, Nauka, Moscow.

[24] A.O.Gelfond and I.M.Kubenskaya, On the theorem of Perron

in the theory of differrence equations (in Russian),

IAN USSR, math. ser., 1953, 17, 2, 83 – 86.

[25] M.A.Evgrafov, New proof of the theorem of Perron (in Russian),IAN USSR, math. ser., 1953, 17, 2, 77 – 82;

[26] G.A.Frejman, On theorems of of Poincaré and Perron (in Russian), UMN, 1957, 12, 3 (75), 243 – 245;

[27] N.E.Nörlund, Differenzenrechnung, Berlin, Springer Verlag, 1924; [28] I.M.Vinogradov, Foundtions of the Number Theory, (in Russian), 1952, GIT-TL;

[29] , J. Diedonne, Foundations of modern analysis, Institut des Hautes Études Scientifiques, Paris, Academic Press, New York and London, 1960

[30] CH.-J. de la Vallée Poussin, Course d’analyse infinitésimale, Russian translation by G.M.Fikhtengolts, GT-TI, 1933;

[31] H.WEYL, Algebraic theory of numbers, 1940, Russian translation by L.I.Kopejkina;

[32] G.Rhin and C. Viola.

On a prmutation group related to ζ(2), Acta Arithmetica 77(1996), 25 – 56;

[33] G.Rhin and C. Viola. The group structure for ζ(3), Acta Arithmetica 97(2001), 269 – 293;

[34] M. Hata. A new irrationality measure forζ(3), Acta Arithmetica 92(2000), 47 – 57;

[35] T.Rivual. La focntion zêta de Riemann prend une infinité valeurs irrationnelles aux entires impairs, C.R.Acad.Sci.Paris, série 1, p. 267 – 270, 2000.

[36] K.Boll, T.Rivual. Irrationalité d‘une infinité valeurs la focntion zêta aux entires impairs, Invent. math., 146, 193 – 207 (2001);

[37] W.Zudilin. One from the nubers ζ(5), ζ(7), ζ(9), ζ(11) is irrational.

(in Russian) UMN 56(4) (2001), 149 – 150;

[38] W.Zudilin. Apéry’s Theorem and problems for the values of ζ-function of Riemann and their q−analogs. (in Russian) Mocow state university, doctoral thesis, Mocow 2004;

[38] L.A.Gutnik, On the decomposition of the difference operators of Poincaré type (in Russian), VINITI, Moscow, 1992, 2468 – 92, 1 – 55;

[39] , On the decomposition of the difference operators of Poincaré type in Banach algebras (in Russian), VINITI, Moscow, 1992, 3443 – 92, 1 – 36;

[40] , On the difference equations of Poincaré type (in Russian), VINITI, Moscow 1993, 443 – B93, 1 – 41;

[41] , On the difference equations of Poincaré type in normed algebras (in Russian), VINITI, Moscow, 1994, 668 – B94, 1 – 44;

[42] , On the decomposition of the difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 2062 – B97, 1 – 41;

[43] , The difference equations of Poincaré type with characteristic polynomial having roots equal to zero (in Russian), VINITI, Moscow, 1997, 2418 – 97, 1 – 20;

[44] , On the behavior of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 3384 – B97, 1 - 41;

[45] , On the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1999, 361 – B99, 1 – 9;

[46] , To the question of the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 2000, 2416 – B00, 1 – 22;

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

[48] , On the Irrationality of Some Quantyties Containing ζ(3) (in Russian), Uspekhi Mat. Nauk, 1979, 34, 3(207), 190;

[49] , On the Irrationality of Some Quantities Containing ζ(3), Eleven papers translated from the Russian, American Mathematical Society, 1988, 140, 45 - 56; [50] , Linear independence over ℚ of dilogarithms at rational points

(in Russian), UMN, 37 (1982), 179-180; english transl. in Russ. Math. surveys 37 (1982), 176-177;

[51] , On a measure of the irrationality of dilogarithms at rational points (in Russian), VINITI, 1984, 4345-84, 1 – 74;

[52] , To the question of the smallness of some linear forms (in Russian), VINITI, 1993, 2413-B93, 1 – 94;

[53] , About linear forms, whose coefficients are logarithms of algebraic numbers (in Russian), VINITI, 1995, 135-B95, 1 – 149;

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

[55] , On the linear forms, whose coefficients are 𝔸 - linear combinations of logarithms of 𝔸 - numbers, VINITI, 1996, 1617-B96, pp. 1 – 23.

[56] , On systems of linear forms, whose coefficients are 𝔸 - linear combinations of logarithms of 𝔸 - numbers, VINITI, 1996, 2663-B96, pp. 1 – 18.

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

[58] , The lower estimate for some linear forms, coefficients of which are proportional to the values of ζ(s) for integer s (in Russian), VINITI, 1997, 3072-B97, 1 – 77;

[59] , On linear forms with coefficients in ℕζ(1 + ℕ)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2000, 3, 1 – 13;

[60] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 1), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2001, 15, 1 – 20;

[61] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 2), Max-Plank-Institut für Mathematik, Bonn, Preprint Series,2001, 104, 1 – 36;

[62] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 57, 1 – 33;

[63] , On the rank over ℚ of some real matrices (in Russian), VINITI, 1984, 5736-84; 1 – 29;

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

[65] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 4), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 142, 1 – 27;

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

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

[68] ,On the difference equation of Poincaré type (Part 1).

Max-Plank-Institut für Mathematik, Bonn,

Preprint Series, 2003, 52, 1 – 44.

[69] , On the dimension of some linear spaces over ℚ, (part 4)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 73, 1 – 38.

[70] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 5), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 83, 1 – 13.

[71] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 6), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 99, 1 – 33.

[72] ,On the difference equation of Poincaré type (Part 2).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 107, 1 – 25.

[73] , On the asymptotic behavior of solutions of difference equation (in English). Chebyshevskij sbornik, 2003, v.4, issue 2, 142 – 153.

[74] , On linear combinations of logarithms of algebraic numbers with algebraic coefficients. International conference ”Diophantine analysis, uniform distributions and applications” August 25-30, 2003, Minsk, Belarus Abstracts pp. 16 – 17.

[75] , On linear forms with coefficients in ℕζ(1 + ℕ), Bonner Mathematishe Schriften Nr. 360, Bonn, 2003, 360.

[76] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 7), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2004, 88, 1 – 27.

[77] ,On the difference equation of Poincaré type (Part 3).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2004, 09, 1 – 34.

[78] , On the dimension of some linear spaces over ℚ, (part 5)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2004,1 – 42.

E-mail: gutnik@gutnik.mccme.ru