Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 160–174. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 165–178. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Zudilin. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

Difference Equations and the Irrationality Measure of Numbers

V. V. Zudilin

Received February, 1997

1. LINEAR DIFFERENCE EQUATIONS

A relation of the form

a_{0η}u_{η+m} + a_{1η}u_{η+m−1} + ⋯ + a_{mη}u_{η} = 0,  η = n, n + 1, n + 2, …  (1)

with given complex numbers a_{jη}, j = 1, …, m, η = n, n + 1, …, is called a linear (homogeneous) equation of order m. Moreover, we assume that a_{0η} ≠ 0 for any integer η ≥ n. By a solution to the difference equation (1) we mean a sequence of numbers u_{n}, u_{n+1}, u_{n+2}, … which satisfies the recursive relation (1).

The solution {u_{η}} to equation (1) is a trivial (or zero) solution if u_{η} = 0 for all integers η ≥ n_{1}. In this case, replacing n by n_{1} in (1), we obtain the usual trivial solution. This is one of the reasons why the change of the subscript η begins with a certain n ∈ ℤ (it is traditionally assumed that n = 0). Two solutions {u_{η}} and {v_{η}} to the difference equation (1) are linearly independent if any one of their linear combinations

{αu_{η} + βv_{η}}, |α| + |β| > 0, α, β ∈ ℂ,

is nontrivial. Otherwise, these solutions are linearly dependent.

If the finite limits

a_{j} = lim_{η→∞} \frac{a_{jη}}{a_{0η}} , j = 1, …, m, a_{m} ≠ 0,

are well defined, then the polynomial φ(λ) = λ^{m} + a_{1}λ^{m−1} + ⋯ + a_{m−1}λ + a_{m} is a characteristic polynomial of the difference equation (1). In the sequel, we shall consider linear difference equations which possess a characteristic polynomial.

The asymptotics of the solution {u_{η}} is closely connected with the roots of the characteristic polynomial. We illustrate this relationship by the assertion which generalizes the classical Poincaré theorem.

Proposition 1 [1, Theorem 1]. Let the sequence u_{n}, u_{n+1}, … be a nontrivial solution to the linear difference equation (1) with the characteristic polynomial φ(λ). Then the limit superior

lim sup_{η→∞} |u_{η}|^{1/η}  (2)

is equal to the modulus of one of the roots of the polynomial φ(λ), and the sequence {u_{η}} is also a solution to the difference equation

u_{η+k} + b_{1η}u_{η+k−1} + ⋯ + b_{kη}u_{η} = 0, η = n, n + 1, n + 2, …,

of order k ≤ m whose characteristic polynomial divides φ(λ) and has all roots whose moduli are equal to (2).

Corollary. Assume that under the conditions of Proposition 1 we have

lim sup_{η→∞} |u_{η}|^{1/η} = |λ_{1}|;  (3)

moreover, let the moduli of the other roots of the characteristic polynomial φ(λ) be different from |λ_{1}|. Then

lim_{η→∞} \frac{u_{η+1}}{u_{η}} = λ_{1}

(in particular, the limit superior in (3) can be replaced by the ordinary one).

Proof. Indeed, according to Proposition 1, the solution {u_{η}} satisfies the linear difference equation

u_{η+1} − b_{1η}u_{η} = 0, η = n, n + 1, n + 2, …,

with the characteristic equation λ−b_{1}, which divides φ(λ). Therefore b_{1} is a root of the polynomial φ(λ), and

lim_{η→∞} \frac{u_{η+1}}{u_{η}} = lim_{η→∞} b_{1η} = b_{1};

whence we have

lim_{η→∞} |u_{η}|^{1/η} = |b_{1}|.

Comparing the last equation to (3) and taking into account that λ_{1} is a unique root of the polynomial φ(λ) whose modulus is equal to |λ_{1}|, we obtain b_{1} = λ_{1}. The assertion is proved.

Proposition 2. Let two sequences {u_{η}} and {v_{η}} be nontrivial solutions to the linear difference equation (1),

lim_{η→∞} \frac{v_{η}}{u_{η}} = α ≠ 0,

and let the following relation hold for the roots λ_{1}, λ_{2}, …, λ_{m} of the characteristic polynomial of this equation:

|λ_{1}| > δ = max_{2≤j≤m}{|λ_{j}|}.  (4)

Then the sequence

r_{η} = u_{η}α − v_{η}, η = n, n + 1, n + 2, …,

which is also a solution to the difference equation (1), satisfies the limit relation

lim sup_{η→∞} |r_{η}|^{1/η} ≤ δ.  (5)

Proof. Assume that inequality (5) does not hold. According to Proposition 1, this is possible only in the case where

lim sup_{η→∞} |r_{η}|^{1/η} = |λ_{1}|.  (6)

In particular, {r_{η}} is a nontrivial solution to equation (1). Since

lim_{η→∞} \frac{r_{η}}{u_{η}} = lim_{η→∞} \frac{u_{η}α − v_{η}}{u_{η}} = α lim_{η→∞} \frac{u_{η}}{u_{η}} − lim_{η→∞} \frac{v_{η}}{u_{η}} = 0,  (7)

we find that

lim sup_{η→∞} |u_{η}|^{1/η} ≥ lim sup_{η→∞} |r_{η}|^{1/η} = |λ_{1}|.

On the other hand, λ_{1} is a root of the characteristic polynomial, maximal in absolute value, and, according to Proposition 1, the last inequality becomes an equality

lim sup_{η→∞} |u_{η}|^{1/η} = |λ_{1}|.  (8)

In order to continue the proof, we need two auxiliary assertions from [2]. We give these assertions in the notation used in our paper.

Lemma 1. If s_{η} ≠ 0, η = n, n + 1, …, and {u_{η}} is a solution to the linear difference equation

u_{η+m} + a_{1η}u_{η+m−1} + ⋯ + a_{mη}u_{η} = 0,  η = n, n + 1, n + 2, …,  (9)

then

v_{η} = \frac{u_{η}}{s_{η}} , η = n, n + 1, n + 2, …,

is a solution to the linear difference equation

v_{η+m} + a′_{1η}v_{η+m−1} + ⋯ + a′_{mη}v_{η} = 0,  η = n, n + 1, n + 2, …,  (10)

a′_{jη} = a_{jη} \frac{s_{η+m−j}}{s_{η+m}} , η = n, n + 1, n + 2, …, j = 1, …, m.

If , in addition,

lim_{η→∞} \frac{s_{η+1}}{s_{η}} = 1,

then the characteristic polynomials of equations (9) and (10) coincide.

Lemma 2. If equation (9) has a solution u_{η}^{(1)} = λ_{1}^{η}, η = n, n + 1, …, then this equation can be written in the form

w_{η+m−1} + b_{1η}w_{η+m−2} + ⋯ + b_{m−1,η}w_{η} = 0,  η = n, n + 1, n + 2, …,  (11)

where

w_{η} = u_{η+1} − λ_{1}u_{η}, η = n, n + 1, n + 2, ….

Moreover, the characteristic polynomials of equations (11) and (9) are related as

(λ − λ_{1})(λ^{m−1} + b_{1}λ^{m−2} + ⋯ + b_{m−1}) = λ^{m} + a_{1}λ^{m−1} + ⋯ + a_{m}.

Let us continue the proof of Proposition 2. The limit relation (8) and the corollary to Proposition 1 allow us to assert that

lim_{η→∞} \frac{u_{η+1}}{u_{η}} = λ_{1}.

Therefore the sequence

s_{η} = \frac{u_{η}}{λ_{1}^{η}} , η = n, n + 1, n + 2, …

satisfies the conditions of Lemma 1, and the difference equation

v_{η+m} + a′_{1η}v_{η+m−1} + ⋯ + a′_{mη}v_{η} = 0,  η = n, n + 1, n + 2, …,  (12)

where

a′_{jη} = \frac{a_{jη}}{a_{0η}} \frac{s_{η+m−j}}{s_{η+m}} , η = n, n + 1, n + 2, …, j = 1, …, m,

has a solution

v_{η}^{(1)} = \frac{u_{η}}{u_{η}/λ_{1}^{η}} = λ_{1}^{η},  η = n, n + 1, n + 2, ….  (13)

By Lemma 2, equation (12) can be represented in the form

w_{η+m−1} + b_{1η}w_{η+m−2} + ⋯ + b_{m−1,η}w_{η} = 0,  η = n, n + 1, n + 2, …,  (14)

where

w_{η} = v_{η+1} − λ_{1}v_{η}, η = n, n + 1, n + 2, ….

The roots of the characteristic polynomial of equation (14) are equal to λ_{2}, …, λ_{m}. We choose a certain basis

{w_{η}^{(1)}}, …, {w_{η}^{(m−1)}}  (15)

in the space of solutions of the linear difference equation (14). According to Proposition 1, we have

lim sup_{η→∞}|w_{η}^{(j)}|^{1/η} ≤ δ,  j = 1, …, m − 1.  (16)

For each j = 2, …, m, we define the solution {v_{η}^{(j)}} to equation (12) in such a way that

v_{η+1}^{(j)} − λ_{1}v_{η}^{(j)} = w_{η}^{(j−1)},  η = n, n + 1, n + 2, ….  (17)

To this end, we set

v_{η}^{(j)} = −\frac{w_{η}^{(j−1)}}{λ_{1}} − \frac{w_{η+1}^{(j−1)}}{λ_{1}^{2}} − \frac{w_{η+2}^{(j−1)}}{λ_{1}^{3}} − ⋯,  η = n, n + 1, n + 2, …, j = 2, …, m.  (18)

The last series converges by (4) and (16). Solutions (18) are linearly independent according to (17) and the linear independence of basis (15). It is easy to show that

lim sup_{η→∞}|v_{η}^{(j)}|^{1/η} = lim sup_{η→∞}|w_{η}^{(j−1)}|^{1/η} ≤ δ,  j = 2, …, m.  (19)

Taking (13) into account, we find that the sequences

{v_{η}^{(1)}}, {v_{η}^{(2)}}, …, {v_{η}^{(m)}}

form a basis in the solution space of equation (12). Therefore the sequences

u_{η}^{(j)} = v_{η}^{(j)}s_{η},  η = n, n + 1, n + 2, …, j = 1, …, m,  (20)

form a basis in the solution space of the initial equation (1). Moreover, {u_{η}^{(1)}} ≡ {u_{η}}, and, according to (19),

lim sup_{η→∞}|u_{η}^{(j)}|^{1/η} = lim sup_{η→∞}|v_{η}^{(j)}|^{1/η} ≤ δ,  j = 2, …, m.  (21)

We express the solution {r_{η}} to the difference equation (1) in terms of the basis (20), i.e.,

r_{η} = ρ_{1}u_{η} + ρ_{2}u_{η}^{(2)} + ⋯ + ρ_{m}u_{η}^{(m)},  η = n, n + 1, n + 2, ….  (22)

According to (21), the relation

lim_{η→∞} \frac{u_{η}^{(j)}}{u_{η}} = 0, j = 2, …, m,

holds; this implies

lim_{η→∞} \frac{r_{η}}{u_{η}} = ρ_{1}.

Comparing the last relation to (7), we infer that ρ_{1} = 0. Therefore resolution (22), together with the limit relations (21), yields

lim sup_{η→∞} |r_{η}|^{1/η} ≤ δ.

However, this contradicts (6). This means that our supposition is not true, and inequality (5) holds.

Proposition 2 is completely proved.

Let us now formulate the general assertion concerning the connection of linear difference equations with the irrationality measure. We will denote by 𝕀 either ℚ or ℚ(i).

Theorem 1. Let {u_{η}} ⊂ 𝕀 and {v_{η}} ⊂ 𝕀 be two linearly independent solutions to the linear difference equation (1),

lim_{η→∞} \frac{v_{η}}{u_{η}} = α ≠ 0,  (23)

and let the natural numbers d_{η}, η = n, n + 1, …, be such that

d_{η}u_{η}, d_{η}v_{η} ∈ ℤ_{𝕀}, η = n, n + 1, n + 2, …,  lim_{η→∞} d_{η}^{1/η} ≤ C,  (24)

where C ≥ 1 is a constant. If λ_{1}, …, λ_{m} are roots of the characteristic polynomial of equation (1), and, moreover, if

|λ_{1}| > δ = max_{2≤j≤m}{|λ_{j}|}

and Cδ < 1, then the number α is irrational, α ∉ 𝕀. Moreover, the irrationality measure of α does not exceed

μ = 1 − \frac{log C + log |λ_{1}|}{log C + log δ},  (25)

i.e., for any ε > 0, there exists q_{∗} = q_{∗}(ε) such that

|α − \frac{p}{q}| ≥ \frac{1}{q^{μ+ε}}

for all p, q ∈ ℤ_{𝕀}, q ≥ q_{∗}.

Proof. The numbers

r_{η} = ũ_{η}α − ṽ_{η},  ũ_{η} = d_{η}u_{η}, ṽ_{η} = d_{η}v_{η},  η = n, n + 1, n + 2, …,  (26)

are approximating linear forms of α and 1 with coefficients in ℤ_{𝕀}. According to Proposition 2 and estimate (24) of the denominators d_{η}, η = n, n + 1, …, we have

lim sup_{η→∞} |r_{η}|^{1/η} ≤ Cδ < 1.  (27)

If α were a rational number, then either Aα − B = 0 or |Aα − B| ≥ 1/D for all A, B ∈ ℤ_{𝕀}, where D ∈ ℕ is the denominator of the number α; in particular, r_{η} = 0 or |r_{η}| ≥ 1/D for all integers η ≥ n. The latter fact contradicts the limit relation (27) and the nontriviality of the sequence {r_{η}} for a sufficiently large η (the solutions {u_{η}} and {v_{η}} are linearly independent). Hence, α ∉ ℤ_{𝕀}; this also implies r_{η} ≠ 0, η = n, n + 1, ….

Each of the linearly independent solutions {u_{η}} and {v_{η}} is nontrivial. According to Proposition 1, we have

lim sup_{η→∞} |u_{η}|^{1/η} = |λ_{j}|

for a certain j, 1 ≤ j ≤ m. If j ≠ 1, then the limit relation

lim sup_{η→∞} |ũ_{η}|^{1/η} ≤ Cδ < 1

holds for the numbers ũ_{η} ∈ ℤ_{𝕀}, η = n, n + 1, …; but this is impossible since |ũ_{η}| ≥ 1 for an infinite set of subscripts η. Therefore j = 1, and by the corollary of Proposition 1 we have

lim_{η→∞} |u_{η}|^{1/η} = |λ_{1}|.

Thus,

lim_{η→∞} |ũ_{η}|^{1/η} = C|λ_{1}|.  (28)

If we now apply the result by G. V. Chudnovsky (see [3, Lemma 3.5]) to relations (27) and (28) for the approximating linear forms (26), then we obtain estimate (25) of the irrationality measure of α.

The theorem is proved.

As an example to Theorem 1, we consider the Apéry difference equation

(η + 1)^{3}u_{η+1} − (34η^{3} + 51η^{2} + 27η + 5)u_{η} + η^{3}u_{η−1} = 0,  η = 1, 2, …,  (29)

and two linearly independent solutions {u_{η}} and {v_{η}} to this equation which are given by the initial conditions

u_{0} = 1, u_{1} = 5, v_{0} = 0, v_{1} = 6.

One can show (see, e.g., [4]) that u_{η}, D_{η}^{3}v_{η} ∈ ℤ, where D_{η} is the least common multiple of the numbers 1, 2, …, η and

lim_{η→∞} \frac{v_{η}}{u_{η}} = ζ(3) = ∑_{k=1}^{∞} \frac{1}{k^{3}} .

The characteristic polynomial λ^{2} − 34λ + 1 = 0 of the difference equation (29) has the roots λ_{1} = (\sqrt{2} + 1)^{4} and λ_{2} = (\sqrt{2} − 1)^{4}. Moreover,

lim_{η→∞} D_{η}^{1/η} = e.

Since (\sqrt{2} − 1)^{4}e^{3} < 1, Theorem 1 yields the irrationality measure

1 − \frac{3 + 4 log(\sqrt{2} + 1)}{3 + 4 log(\sqrt{2} − 1)} = 13.4178202 …

of the number ζ(3).

2. THE IRRATIONALITY MEASURE

As an application of Theorem 1, we give an example of a linear difference equation whose characteristic polynomial is

(λ−λ_{1}) ⋯ (λ−λ_{m}) = λ^{m}+a_{1}λ^{m−1}+⋯+a_{m−1}λ+a_{m},  a_{j} ∈ ℚ, j = 1, …, m, a_{m} ≠ 0. (30)

The root λ_{1} which is the largest in absolute value is assumed to be real and positive, i.e.,

λ_{1} = |λ_{1}| > δ = max_{2≤j≤m}{|λ_{j}|}.  (31)

We set

τ(z) = (1 − λ_{1}z) ⋯ (1 − λ_{m}z) = 1 + a_{1}z + ⋯ + a_{m}z^{m},  a_{j} ∈ ℚ, j = 1, …, m, a_{m} ≠ 0. (32)

The generating function

U(z) = τ(z)^{−s} = ∑_{ν=0}^{∞} u_{ν}z^{ν},  s ∈ (0, 1) ∩ ℚ,  (33)

satisfies the linear differential equation τy′ + sτ′y = 0. (We take a branch of function (33) which assumes positive values on the cut [0, 1/λ_{1}).) Thus,

(1 + ∑_{μ=1}^{m} a_{μ}z^{μ}) × ∑_{ν=0}^{∞}(ν + 1)u_{ν+1}z^{ν} + s ∑_{μ=1}^{m} μa_{μ}z^{μ+1} × ∑_{ν=0}^{∞} u_{ν}z^{ν} = 0

or

∑_{η=0}^{∞}((η + 1)u_{η+1} + ∑_{μ=1}^{m} a_{μ}(η + 1 + (s − 1)μ)u_{η+1−μ})z^{η} = 0,

where u_{0} = 1 and u_{−m+1} = ⋯ = u_{−2} = u_{−1} = 0.

Thus, the sequence of numbers {u_{η}} satisfies the difference equation

(η + 1)u_{η+1} + ∑_{μ=1}^{m} a_{μ}(η + 1 + (s − 1)μ)u_{η+1−μ} = 0,  η = 0, 1, 2, …,  (34)

whose characteristic polynomial is (30).

Let now κ(z) ∈ 𝕀[z] be a nonzero polynomial, and let the function V (z) satisfy the differential equation

τy′ + sτ′y = κ  (35)

with the initial condition V (0) = 0. This condition and the relation

\frac{d}{dz} (τ(z)^{s}V (z)) = τ(z)^{s−1}(τ(z)V′(z) + sτ′(z)V (z)) = τ(z)^{s−1}κ(z)

yield

V (z) = τ(z)^{−s} ∫_{0}^{z} τ(x)^{s−1}κ(x) dx = ∑_{η=0}^{∞} v_{η}z^{η}.  (36)

In terms of the generating function, the differential equation (35) can be rewritten in the form

∑_{η=0}^{∞}((η + 1)v_{η+1} + ∑_{μ=1}^{m} a_{μ}(η + 1 + (s − 1)μ)v_{η+1−μ})z^{η} = κ(z),

where we set v_{−m+1} = ⋯ = v_{−1} = v_{0} = 0. Thus, beginning with η = deg κ, the sequence {v_{η}} is a solution to the difference equation (34).

If the solutions {u_{η}} and {v_{η}} to equation (34) were linearly dependent, then αU(z)−V (z)=π(z) would hold for certain α ∈ ℂ and π(z) ∈ ℂ[z]. After the multiplication of both sides of this equation by τ(z)^{s} and after the differentiation, we find that τ(z)^{s−1}κ(z) = (τ(z)^{s}π(z))′, i.e., π(z) is not a solution to the differential equation (35). Thus, the requirement that equation (35) should have no polynomial solution for a chosen κ(z) entails the linear independence of the solutions {u_{η}} and {v_{η}}.

Lemma 3. Let the polynomial τ(z), (32), be chosen in accordance with conditions (31), κ(z) be an arbitrary polynomial, and let the functions U(z) and V (z) be defined by relations (33) and (36), respectively. Then

lim_{η→∞} \frac{v_{η}}{u_{η}} = ∫_{0}^{1/λ_{1}}τ(x)^{s−1}κ(x) dx.

Proof. By virtue of (31), the point z = 1/λ_{1} is the only singularity of the functions U(z) and V (z) in the disc |z| ≤ r, where 1/λ_{1} < r < 1/δ. Hence, by the Cauchy formula, we have

u_{η} = \frac{1}{2πi} ∫_{K} \frac{U(z)}{z^{η+1}} dz, v_{η} = \frac{1}{2πi} ∫_{K} \frac{V (z)}{z^{η+1}} dz,  η = 0, 1, 2, …,  (37)

where the contour of integration K is the boundary of the disc |z| ≤ r with the radial cut (1/λ_{1}, r) (by the condition s < 1, we can also integrate over the path passing through the point z = 1/λ_{1}).

The functions (1 − λ_{1}z)^{s}U(z) and (1 − λ_{1}z)^{s}V (z) are analytic in the disc |z| ≤ r, and therefore, in the smaller disc |λ_{1}z − 1| ≤ λ_{1}r − 1. Therefore these functions can be represented in the form

U(z) = (1 − λ_{1}z)^{−s}Ũ(λ_{1}z − 1), V (z) = (1 − λ_{1}z)^{−s}Ṽ(λ_{1}z − 1),

where the functions Ũ(t) and Ṽ(t) are analytic in the disc |t| ≤ λ_{1}r − 1. Moreover, we have

\begin{matrix}Ũ(0) = ((1 − λ_{1}z)^{s}U(z))|_{z=1/λ_{1}} = ∏_{j=2}^{m}(1 − \frac{λ_{j}}{λ_{1}} )^{−s}, \\ Ṽ(0) = ((1 − λ_{1}z)^{s}V (z))|_{z=1/λ_{1}} = ∏_{j=2}^{m}(1 − \frac{λ_{j}}{λ_{1}} )^{−s} ∫_{0}^{1/λ_{1}}τ(x)^{s−1}κ(x) dx \end{matrix}  (38)

for these functions.

We pass now to formulas (37) for the function U(z). We have

U(z) = e^{πis}(λ_{1}z − 1)^{−s}Ũ(λ_{1}z − 1)

on the upper bank of the cut, and

U(z) = e^{−πis}(λ_{1}z − 1)^{−s}Ũ(λ_{1}z − 1)

on the lower one. The integral (37) is equal to O(r^{−η}) on the circle |z| = r; therefore,

u_{η} = \frac{e^{πis} − e^{−πis}}{2πi} ∫_{1/λ_{1}}^{r} (λ_{1}z − 1)^{−s}Ũ(λ_{1}z − 1)\frac{dz}{z^{η+1}} + O(r^{−η})

= \frac{sin πs}{π} ∫_{1/λ_{1}}^{r} (λ_{1}z − 1)^{−s}Ũ(λ_{1}z − 1)\frac{dz}{z^{η+1}} + O(r^{−η}).

After the change of the variable λ_{1}z − 1 = t, the integral in the last expression reduces to the form

∫_{1/λ_{1}}^{r} (λ_{1}z − 1)^{−s}Ũ(λ_{1}z − 1)\frac{dz}{z^{η+1}} = ∫_{0}^{λ_{1}r−1}t^{−s}Ũ(t)\frac{λ_{1}^{η} dt}{(1 + t)^{η+1}} = λ_{1}^{η} ∫_{0}^{λ_{1}r−1}t^{−s}Ũ(t)e^{−(η+1) log(1+t)} dt

= λ_{1}^{η} ∫_{0}^{λ_{1}r−1}t^{−s}Ũ(t)e^{−(η+1)t(1+O(t))} dt.

Theorem 1.2.1 in [5] yields the asymptotic estimate

∫_{0}^{λ_{1}r−1}t^{−s}Ũ(t)e^{−(η+1)t(1+O(t))} dt ∼ Ũ(0)Γ(1 − s)η^{s−1} as η → ∞.

Hence,

u_{η} ∼ \frac{sin πs}{π}λ_{1}^{η}Ũ(0)Γ(1 − s)η^{s−1} as η → ∞.

By analogy, we find that

v_{η} ∼ \frac{sin πs}{π}λ_{1}^{η}Ṽ(0)Γ(1 − s)η^{s−1} as η → ∞.

Therefore

lim_{η→∞} \frac{v_{η}}{u_{η}} = \frac{Ṽ(0)}{Ũ(0)},

and the substitution of values (38) yields the required result.

The lemma is proved.

Remark. The arguments used above can be easily extended to the case of negative λ_{1}. Therefore condition (31) can be replaced by

λ_{1} ∈ ℝ,  |λ_{1}| > δ = max_{2≤j≤m}{|λ_{j}|}.  (39)

Now we estimate the denominators of the sequences {u_{η}} and {v_{η}}.

Lemma 4. Let us define natural numbers A, B, and C as follows: B is the denominator of the rational number s ∈ (0, 1); A is the least common denominator of the numbers a_{j}/B, j = 1, …, m; C is the least common denominator of the coefficients of the polynomial κ(z) ∈ 𝕀[z]. Then, for the sequence

d_{η} = A^{η} × C × D_{η} × ∏_{p|B} p^{[\frac{η}{p−1}]},  η = 0, 1, 2, …,  (40)

and for the functions U(z) and V (z) which are chosen in accordance with (33) and (36) we have

d_{η}u_{η}, d_{η}v_{η} ∈ ℤ_{𝕀}, η = 0, 1, 2, …

(as before, D_{η} denotes the least common multiple of the numbers 1, 2, …, η.)

Proof. By the Taylor formula, we have

U(z) = (1 + a_{1}z + ⋯ + a_{m}z^{m})^{−s} = 1 + ∑_{ν=1}^{∞}(−1)^{ν} \frac{s(s + 1) ⋯ (s + ν − 1)}{ν!}(a_{1}z + ⋯ + a_{m}z^{m})^{ν}.

If s = B_{0}/B is an irreducible fraction and the prime number p does not divide B, then this number enters into each rational number

\frac{s(s + 1) ⋯ (s + ν − 1)}{ν!} = \frac{1}{B^{ν}} \frac{B_{0}(B_{0} + B) ⋯ (B_{0} + (ν − 1)B)}{ν!},  ν = 0, 1, 2, …, (41)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

in a nonnegative power (see, e.g., [6, Ch. I, Appendix]). Hence, the denominators of numbers (41) contain only prime numbers which divide B. Any prime number p of this kind enters into the first factor on the right-hand side of (41) in the power of ν and in the power not exceeding

[\frac{ν}{p} ] + [\frac{ν}{p^{2}} ] + [\frac{ν}{p^{3}} ] + ⋯ ≤ ν(\frac{1}{p} + \frac{1}{p^{2}} + \frac{1}{p^{3}} + ⋯) = \frac{ν}{p − 1}  (42)

the second factor, ν = 0, 1, 2, …. Thus, when the νth term of the sum

U(z) = 1 + ∑_{ν=1}^{∞}(−1)^{ν} \frac{B_{0}(B_{0} + B) ⋯ (B_{0} + (ν − 1)B)}{ν!}(\frac{a_{1}}{B} z + ⋯ + \frac{a_{m}}{B} z^{m})^{ν}

is multiplied by the number

f_{ν} = A^{ν} × ∏_{p|B} p^{[\frac{η}{p−1}]},

this term becomes a polynomial with integer coefficients. Therefore the coefficient u_{η} in decomposition (33) becomes an integral after the multiplication by f_{η}, η = 0, 1, 2, …. Similar arguments allow us to assert that the coefficient w_{η} in the decomposition

τ(z)^{s−1}κ(z) = ∑_{η=0}^{∞} w_{η}z^{η}

becomes an integer in ℤ_{𝕀} after multiplication by Cf_{η}, η = 0, 1, 2, …, since Cκ(z) ∈ ℤ_{𝕀}[z]. According to (36), we have

V (z) = τ(z)^{−s} ∫_{0}^{z} τ(x)^{s−1}κ(x) dx = ∑_{η=0}^{∞} u_{η}z^{η} × ∑_{η=1}^{∞} \frac{w_{η}}{η} z^{η},

and therefore the multiplication of v_{η} by d_{η} yields a number from ℤ_{𝕀}, η = 0, 1, 2, …. The lemma is proved.

In the case where the denominator of the number s is a power of 2, the assertion of Lemma 4 can be strengthened.

Lemma 5. Let B = 2^{k}, k ∈ ℕ, A be the least common denominator of the numbers a_{j}/(2B), j = 1, …, m, and let C be the least common denominator of the coefficients of the polynomial κ(z) ∈ 𝕀[z]. Then, for the sequence

d_{η} = A^{η} × C × D_{η},  η = 0, 1, 2, …,  (43)

and for the functions U(z) and V (z) which are chosen in accordance with (33) and (36) we have

d_{η}u_{η}, d_{η}v_{η} ∈ ℤ_{𝕀}, η = 0, 1, 2, ….

Proof. According to (42), the prime number 2 enters into ν! in the power not higher than ν, ν = 0, 1, …. Therefore the multiplication of the νth term of the decomposition

U(z) = 1 + ∑_{ν=1}^{∞}(−1)^{ν} \frac{2^{ν}B_{0}(B_{0} + B) ⋯ (B_{0} + (ν − 1)B)}{ν!}(\frac{a_{1}}{2B} z + ⋯ + \frac{a_{m}}{2B} z^{m})^{ν}

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

by f_{ν} = A^{ν} yields a polynomial with integral coefficients. The remaining part of the proof repeats the proof of the preceding lemma with the use of the new value of f_{ν}.

Let us summarize the results of this section in the following assertion.

Theorem 2. Let the polynomial τ(z) ∈ ℚ[z] defined by relation (32) satisfy condition (39), s ∈ (0, 1) be a rational number with denominator B, and let the polynomial κ(z) ∈ 𝕀[z] be chosen such that any solution to the differential equation (35) is not a polynomial. Let A be the least common denominator of the numbers a_{j}/(2B) in the case where B = 2^{k}, k ∈ ℕ, and of the numbers a_{j}/B for other B, j = 1, …, m, and let

C = \{ \begin{matrix}Ae & if B = 2^{k}, k ∈ ℕ \\ A exp\{1 + ∑_{p|B} \frac{ln p}{p − 1}\} & in other cases. \end{matrix}  (44)

If Cδ < 1, then the irrationality measure of

α = ∫_{0}^{1/λ_{1}}τ(z)^{s−1}κ(z) dz

does not exceed

μ = 1 − \frac{log C + log |λ_{1}|}{log C + log δ}.

Proof. The sequences {u_{η}} and {v_{η}}, which are defined by the generating functions (33) and (36), satisfy the linear difference equation (34) and are linearly independent. In addition, by Lemma 3, we have

lim_{η→∞} \frac{v_{η}}{u_{η}} = α

and {d_{η}u_{η}}, {d_{η}v_{η}} ∈ ℤ_{𝕀} according to Lemmas 4 and 5; the sequence {d_{η}} is chosen in accordance with (40) or (43). Since

lim_{η→∞} D_{η}^{1/η} = e,  lim_{η→∞}(∏_{p|B} p^{[\frac{η}{p−1}]})^{1/η} = ∏_{p|B} p^{\frac{1}{p−1}} ,

we obtain

lim_{η→∞} d_{η}^{1/η} = C.

The application of Theorem 1 completes the proof of the theorem.

3. THE CHOICE OF THE CHARACTERISTIC POLYNOMIAL

The aim of this section is to present the characteristic polynomial φ(λ) ∈ ℚ[λ] all of whose roots, except for one, lie in the closed disc |λ| ≤ δ with a sufficiently small δ.

Lemma 6. Let m ∈ ℕ, m ≥ 3, and let the real numbers a and b satisfy the conditions

|a| ≥ 1,  |b| ≥ 5,  |\frac{a}{b}| < \frac{1}{2} sin^{m−1} \frac{π}{m − 1}.  (45)

Then the root λ_{1} of the polynomial φ(λ) = λ^{m} − bλ^{m−1} + a which is maximal in absolute value is real and sign λ_{1} = sign b; the other roots λ_{2}, …, λ_{m} lie in the disc |λ| < r + 2^{1/(m−1)}r^{2}, where r = |a/b|^{1/(m−1)}.

Proof. According to (45), the inequality |b|/2 > |a| ≥ 1 holds. Hence, (|b|/2)^{m} > |a|, and the number

φ(\frac{b}{2}) = −(\frac{b}{2})^{m} + a

has the same sign as b^{m}. Therefore, at least one root of the polynomial φ(λ) lies in the interval

(−∞, \frac{b}{2})  for b < 0,  (\frac{b}{2}, +∞)  for b > 0.  (46)

Let us denote by λ_{1} one of these roots (below we shall show that this root is unique). Since m ≥ 3,

0 < r = |\frac{a}{b}|^{1/(m−1)} < \frac{1}{2^{1/(m−1)}} sin \frac{π}{m − 1} ≤ \frac{1}{\sqrt{2}}

and

\frac{1}{\sqrt{2}} \frac{1}{|a|} + \sqrt{2}\frac{1}{|b|} ≤ \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{5} = \frac{7\sqrt{2}}{10} < 1

for |a| ≥ 1 and |b| ≥ 5, we obtain

|λ_{1}| > \frac{|b|}{2} > \sqrt{2}(\frac{|b|}{2|a|} + 1) > r(\frac{|b|}{2|a|} + 1).  (47)

Consider the auxiliary polynomial

ψ(λ) = (λ^{m−1} − \frac{a}{b})(λ − b) = λ^{m} − bλ^{m−1} − \frac{a}{b}λ + a.

One of its roots is equal to λ′_{1} = b and the other roots λ′_{2}, …, λ′_{m} lie at the vertices of the regular (m − 1)-gon on the circle |λ| = r.

We fix some roots λ′_{j}, j = 2, …, m, of the polynomial ψ(λ) and denote

ρ = min_{2≤l≤m}{|λ′_{j} − λ_{l}|}.

Since

φ(λ) − ψ(λ) = \frac{a}{b}λ,

we find for λ = λ′_{j} that

(λ_{1} − λ′_{j})(λ_{2} − λ′_{j}) ⋯ (λ_{m} − λ′_{j}) = φ(λ′_{j}) = \frac{a}{b}λ′_{j};

this implies

r^{m} = |\frac{a}{b}| × |λ′_{j}| = |φ(λ′_{j})| ≥ |λ_{1} − λ′_{j}|ρ^{m−1} ≥ (|λ_{1}| − r)ρ^{m−1} > r \frac{|b|}{2|a|}ρ^{m−1}

according to (47). The final result is

ρ < r(\frac{2|a|}{|b|})^{1/(m−1)} = 2^{1/(m−1)}r^{2} = ε.

We have found out that in the ε-neighborhood of any root λ′_{j}, j = 2, …, m, of the polynomial ψ(λ), there is a certain root λ_{j} of the polynomial φ(λ). These ε-neighborhoods are mutually disjoint and do not intersect interval (46), since

|λ′_{j} − λ′_{l}| ≥ r|e^{2πi/(m−1)} − 1| = 2r sin \frac{π}{m − 1} > 2r(2\frac{|a|}{|b|})^{1/(m−1)} = 2ε

according to (45). It remains to note that

|λ_{j}| ≤ |λ′_{j}| + ρ < r + ε, j = 2, …, m,

and the root λ_{1} is a root of the polynomial φ(λ) maximal in the absolute value. The lemma is proved.

Remark. Actually, the boundary on the roots λ_{2}, …, λ_{m} of the polynomial

φ(λ) = λ^{m} − bλ^{m−1} + a

can be considerably refined with a specific choice of the numbers m, a, and b. We do not pose this problem here since Lemma 6 makes it possible to apply Theorem 2.

Let now τ(z) = az^{m} − bz + 1, m ≥ 3, and let κ(z) be a nonzero polynomial. Assume that the polynomial π(z) of degree k ≥ 0 is a solution to equation (35). Then π(z) = cz^{k} + ⋯, c ≠ 0, and

κ(z) = τ(z)π′(z) + sτ′(z)π(z) = ac(k + sm)z^{m+k−1} + ⋯;

this implies that deg κ = m + k − 1 ≥ m − 1. Hence, if we assume additionally that deg κ < m − 1, then any solution to equation (35) is not a polynomial.

The following assertion is a corollary of Theorem 2 and Lemma 6.

Theorem 3. Let m ∈ ℕ, m ≥ 3, and let the rational numbers a and b satisfy conditions (45). Assume that B is the denominator of the rational number s ∈ (0, 1), A is the least common denominator of the numbers a/(2B) and b/(2B) in the case B = 2^{k}, k ∈ ℕ, and of the numbers a/B and b/B for other B, the constant C is chosen in accordance with (44), and

δ = |\frac{a}{b}|^{1/(m−1)} + 2^{1/(m−1)}|\frac{a}{b}|^{2/(m−1)}.

If Cδ < 1 and the degree of the polynomial κ(z) ∈ 𝕀[z] is lower than m − 1, then the irrationality measure of the number

∫_{0}^{z_{1}} \frac{κ(z)}{(az^{m} − bz + 1)^{s}} dz,

where z_{1} is a root of the polynomial az^{m} − bz + 1 which is minimal in the absolute value, does not exceed the value

μ = 1 − \frac{log C − log |z_{1}|}{log C + log δ}.

In [7], a special case (m = 3, s = 1/2, a = 4, b is a multiple of 4) of Theorem 3 was considered. The exact localization of the roots of the characteristic polynomial φ(λ) in this case allows one to obtain the irrationality measure of values of certain elliptic integral with a high degree of accuracy.

ACKNOWLEDGMENTS

This work was partially supported by the Russian Foundation for Fundamental Research (project 97-01-00181).

REFERENCES

1. Buslaev, V. I., Relations for Coefficients and Singular Points of a Function, Mat. Sb., 1986, vol. 131 (173), no. 3 (11), pp. 357–384.

2. Evgrafov, M. A., A New Proof of the Perron Theorem, Izv. Akad. Nauk SSSR, Ser. Mat., 1953, vol. 17, no. 2, pp. 77–82.

3. Chudnovsky, G. V., Hermite–Padé Approximations to Exponential Functions and Elementary Estimates of the Measure of Irrationality of π, Lect. Notes Math., 1982, vol. 925, pp. 299–322.

4. Beukers F., A Note on the Irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., 1979, vol. 11, pp. 268–272.

5. Evgrafov, M. A., Asymptoticheskie otsenki i tselye funktsii (Asymptotic Estimates and Entire Functions), 3rd ed., Moscow: Nauka, 1979.

6. André, Y., G-Functions and Geometry, Aspects of Mathematics, vol. E13, Braunschweig: Vieweg, 1989.

7. Zudilin, V. V., Recurrence Sequences and Measure of Irrationality for Values of Elliptic Integrals, Mat. Zametki, 1997, vol. 61, no. 5, pp. 785–789.

Translated by S. Vakhromeev