1. Introduction
Ever since Hippasus of Metapontum made the astonishing discovery of the existence of irrational numbers, the irrationality or transcendence of special values has attracted much attention. From the simplistic proof for the irrationality of and e, mathematicians have developed a variety of different measures to prove the irrationality of special values, as well as describe its irrationality in terms of rational approximations.
The French mathematician Charles Hermite first came up with the idea of using rational functions to approximate and prove the irrationality or transcendence of values of exponential functions at rational points. This method of approximation by rational functions is called Padé approximation. This method motivated many mathematicians to study Diophantine approximations, for example, Maier [
1], Siegel [
2] and Chudnovsky [
3].
One of the most memorable and significant yet surprising findings in the irrationality of values is Apéry’s proof of
’s irrationality in 1978. The original sketch of proof published by Apéry involves two novel series
and
named Apéry numbers and uses their quotients to approach the value of
(See [
4]), which was once considered miraculous and unexpected.
However, such series
and
for many other Diophantine approximation problems naturally appear in Padé approximations or Padé-type approximations, especially in the study of generalized hypergeometric functions. A number of significant works have been conducted by Maier [
1], followed by Chudnovsky [
3], who sketched a general form of Padé approximants to generalized hypergeometric functions.
In this paper, we will follow the idea of using Padé or partial Padé approximants to study the irrationality of values of special functions. We will first recall the result of irrationality proof of the exponential values at rational points, and then study the irrationality property of some special types of generalized hypergeometric functions, construct the Maier–Chudnovsky type approximations and derive upper bounds for the irrationality measures. More specifically, we will derive the following main theorem
Theorem 1. For the following special formal power series,where are positive integers such that and there exists an integer in that is co-prime with b. Then, there is a nonzero polynomial depending on ; if is not a root of , then is irrational and the irrationality measure is bounded by:See Proposition 5 for the explicit form of . A detailed definition of irrationality measure can be found in Definition 1. This concept is also called the Liouville–Roth constant or irrationality exponent. It is used to measure how well an irrational real number can be approximated by rational numbers. Finding the values or bounds of irrationality measure is a very important topic in Diophantine approximation. For example, the irrationality measure for irrational algebraic numbers is the Fields Medal-winning work of Roth. The main contribution in this paper is the bounds for irrationality measure for values of the function at nonzero rational numbers x. Special cases of the functions are summarized below.
Example 1. The form of formal power series in Theorem 1 includes the following important functions upon choosing different parameters:
- 1.
When irrational at nonzero irrational points was studied by Hermite [5]. In this degenerate case, the irrationality measure conclusion still holds; in other words, we have for nonzero rational number x. - 2.
When is related to the exponential integral bywith γ being the Euler–Mascheroni constant. The irrationality of was studied by Maier [1]. - 3.
When is related to the Gauss error function via the followingThis error function is widely used in probability, statistics and PDE. - 4.
When or , the function represents a large class of Kummer’s confluent hypergeometric functions (also known as confluent hypergeometric function of the first kind), which is the solution of Kummer’s differential equations.
The main ingredients in the proof Theorem 1 are the following three steps:
The first step is to use a combinatorial cancellation lemma (Lemma 2) to construct partial Padé approximations to . The key idea here is to sacrifice the order of remainder functions to obtain control of denominators of coefficients in the polynomials.
The second step is to use a mod-p method to prove the non-vanishing of the remainder terms and thus implies the irrationality of those values.
The last step is to apply Tannery’s theorem to estimate the polynomials and remainder terms (Lemmas 5 and 6), and obtain the upper bound for irrationality measure based on a folklore lemma (Lemma 1).
The paper is organized as follows. In
Section 2, we review some basic background including Padé approximations and irrationality measures, and the folklore lemma (Lemma 1) on bounds of irrationality measures. In
Section 3, we introduce the combinatorial cancellation lemma (Lemma 2) and reconstruct the Padé approximation of exponential functions. This also leads to precise estimates of denominators and remainders, and hence the exact value of the irrationality measure of
at nonzero rational
x. In
Section 4, we construct explicit partial Padé-type approximants to general
. In
Section 5, we prove the non-vanishing nature of the remainder via a modulo-
p argument similar to Maier [
1]. In
Section 6, we obtain estimates of denominators and remainders, hence the proof of the main theorem. In
Section 8, we point out some rational values for parameters excluded from the theorem and study the differential systems satisfied by the function. In
Section 9, we discuss the recent important results of Fischler and Rivoal about irrationality measures on values of Type-
E functions.
3. Combinatorial Cancellation Lemma and Padé Approximation
From Equation (
1), we need enough vanishing for coefficients in the expansion of
to obtain explicit partial Padé expansions. An important ingredient in those constructions for hypergeometric functions is the following combinatorial cancellation lemma of binomial numbers.
Lemma 2 (Cancellation lemma)
. Let be a polynomial of degree strictly less than n, then Proof. We first prove the case of
with
. Define the logarithmic differential operator
. Notice that
By the linearity of
S in the left-hand side of the equality, we conclude the proof of the cancellation lemma. □
The lemma was used by Maier [
1] and later Chudnovsky [
3] to construct explicit forms of partial Padé approximations.
It is worth mentioning that there are also interesting developments using the combinatorical properties of binomial coefficients to approximate certain operators, see [
12,
13,
14,
15].
3.1. -Padé Approximants of
In order to show the power of the cancellation lemma, we find -Padé approximants of by this lemma, and use Padé approximants to find the exact value of the irrationality measure of at nonzero rational number x. This can be viewed as a degenerate case of our main Theorem 1 with . More specifically, we will show the following result by Lemmas 1 and 2.
Proposition 1. The irrationality measure of with nonzero rational number x is 2.
Hermite first proved the irrationality of
and Siegel [
2] later interpreted the proof using Padé approximation of exponential function. The proof by Siegel uses the appropriate differential operator acting on both sides of equation (
1). Here, we approach the construction by Lemma 2.
Proposition 2 (Padé approximation of
)
. Let be a pair of positive integers. Define polynomials of degree n and of degree m byThen . Proof. Using the Taylor expansion of
, we have
The summation above for
forms
. So we only need to prove that
For
k such that
, we have
This expression is a polynomial of
i with degree
, which is strictly less than
n. So by Lemma 2, we have
Hence we have the conclusion. □
This proposition also gives us the explicit form of the remainder term
as follows
Here,
is the confluent hypergeometric function and it has the integration form
Plugging this into the remainder term, we obtain
3.2. Irrationality Measure of
In this section, we calculate the irrationality measure of at nonzero rational x.
Proposition 3. The irrationality measure of for is equal to 2.
Proof. For positive integers
, we have the following integral representation for
from the expansion formula above:
When
n is large enough, we will always have
We take
, the growth rate of
is given by
.
Now we estimate the remainder term:
The factor
could be neglected as it gives a factor not related to
n.
Using trigonometric substitution
:
So
. Hence,
. Notice that the coefficients in
and
are integers. When
with
p and
q nonzero integers, we need to multiply
on both sides of
and we still have the following inequality for
n large enough
with any
fixed. So Lemma 1 implies that the irrationality measure of
. □
More generally, the precise estimates above helped Davis to obtain the following.
Theorem 2 (Davis [
16])
. For any , there exists an infinite sequence of rational numbers such thatThe constant is not improvable. 4. Partial Padé Approximation
In this section, we study the Partial Padé approximation to a special function defined by:
where
are positive integers,
. We use a similar method of explicitly constructing the Padé approximants, and we generalize the “mod-p” proof for the non-vanishing nature of the remainder terms as Maier [
1], to complete the proof of irrationality for this type of function. Two special choices of parameters
give two different types of confluent hypergeometric functions. Notice that when
,
When
,
We will first briefly introduce similar work by Maier. Historically, in Maier’s original paper [
1], he provided the proof for the irrationality of a similar function
:
His construction of Padé approximants was complicated as it involves differential operators of two variables (see [
1]). With Lemma 2, however, we can directly construct an explicit form of Padé approximants for this kind of function.
We will first construct a general form of Padé approximants to
:
Unlike the case for exponential functions, the coefficients in the exact Padé approximation are no longer integers, and the least common multiples of denominators may grow rapidly. The vanishing order of the remainder terms is not enough to beat the growth rate to apply Lemma 1 to bound the irrationality measure. So the basic idea is to sacrifice the vanishing order of
to compensate the integrality of the coefficients in
and
.
Proposition 4. Let be two positive integers such that , andthe least common multiple of those numbers inside. We define polynomials byThen is . Proof. Similar to the exponential function, we have
where
When
, we have
The term
is a degree-
polynomial of
i and
is a degree-
polynomial of
i. So the total degree of
i is
, which is less than
n. So lemma 2 implies
for all
. Let
Next we check
. The coefficients of
are
Since
, we have
From the choice of
d, we have
So
for
.
In conclusion, we have
and the remainder term
is
. □
More explicitly, the remainder term has the following form
5. Nonvanishing of Remainder Term
In the irrationality proof of
for
, the remainder term is nonzero by the integral formula (3). In our construction of partial Padé approximants, the remainder term does not have a convenient integration formula to show the non-vanishing nature of the remainder term. In this section, we apply a mod-
p argument for infinitely many primes
p similar to Maier [
1].
In the following discussion, we assume . Let be the minimal integer such that and . In addition, we let
From the choice of
and
w, we have
. Since
, there are infinitely many prime numbers
p such that
is an integer, by the Dirichlet prime number theorem. We choose such prime numbers
p and let
where
a is a fixed positive integer such that
.
Let
be the
p-adic valuation of nonzero rational numbers at prime number
p. We use the convention that
for convenience. We have the following results for the
p-adic valuation of
d.
Lemma 3. Under the choice of in (6), when p is large enough, the p-adic valuation of integer d is . Proof. From the choice of p, we have . For a large enough p, there exists or such that . We have and . So . Since , so and . □
We need Lucas’s theorem about the results of the binomial coefficient modulo p.
Theorem 3 (Lucas [
17])
. Let p be a prime number. Assume and are base p expansions of non-negative integers y and z, respectively. Thenwith the convention that if . Definition 2. We use the (rising) Pochhammer symbol defined by We have the following proposition on the non-vanishing of modulo p for nonzero rational numbers .
Proposition 5. Assume . Let and be a nonzero degree-c rational polynomial defined bywhere if and if . Let be a nonzero rational number that . Then, for a large enough p such that and are chosen in (6), the rational number modulo p, or in other words, and . Proof. We choose
p large enough so that
. We show that most of the coefficients of
are multiples of
p. The coefficient
is the product of the following integer factors
Now we assume integer
modulo
p, in other words, each of the factors is not zero modulo
p, and look for all the possible pairs
. If the last factor is not zero modulo
p, then we have
for positive integers
such that
and
. Since
So
. We consider the following two cases:
If
, then
, so
. Since the third factor
modulo
p, we have
. So, the first factor
modulo
p implies
. So
. In this case,
and
. So
and
. The second factor
modulo
p because
. Consider the fourth factor
. Since
from the choice of
a, and
, we have
from Theorem 3. So
, in other words,
. When
, then
is an integer and
and
. When
, then
is not an integer for large
p, and
. So in either case, we have
. Plugging this into
, we have
where
.
If
, then we have
. So
and the third factor is nonzero modulo
p. Since
, we have
. So the first factor being nonzero modulo
p implies that
. Notice that if we replace
i by
, then all the binomial factors are the same modulo
p. So the argument in the first case implies that
and
where
.
Denote by
. In the two cases above, we have
and the product of the three terms is independent of
k. We denote the product by
M and it is nonzero modulo
p. Let
, then
, and third factor is
The right-hand side only depends on
and we denote it by
. The last factor is congruent to
We denote the factor
since it also depends on
. In conclusion, we have.
The summation on the right-hand side is equal to
Since
, this term is not equal to zero modulo
p when
p is large enough. So we have
modulo large prime number
p. □
In conclusion, when
, we have infinitely prime numbers
p such that
as long as
is not a root of polynomial
. If
is a rational number, this implies that
. Next we show that
as
. Then, it contradicts
being rational. In fact, we show more about the growth rate of
and
to obtain bounds of the irrationality measure of
.
6. Estimates of Growth Rate
In this section, we carry out the estimate of and . First we fix a positive integer. Let and m go to infinity.
Lemma 4 (Estimate of d). There is a positive number C such that for large enough.
Proof. Let
be any positive real number. From the prime number theorem, it is well known that
for
k large enough; for example, see [
4] (Footnote 5). From the definition of
d in Proposition 4, we have
So
. In conclusion, there is
such that
for large
n. □
Lemma 5 (Estimate of
)
. For any and , the following inequalityholds for large enough. Proof. Recall the expression of
,
Notice that the binomial numbers have bounds
and
So
for some fixed positive number
C. So the lemma is proved. □
Lemma 6 (Estimate of
)
. For any , the following inequalityholds for large enough. Proof. We rewrite
as
For a fixed non-negative integer
k, we have
Next we study the behaviour of
as
for each fixed
k. We rewrite it as
where
,
,
and
are constants independent of
k and
m. By Stirling’s formula, we have
Apply the expansion of
Then we obtain
where constants
do not depend on
.
In other words, there are constants
such that
Applying Tannery’s theorem to the summation formula of
, we have
Applying Stirling’s formula to the denominator, we obtain the growth rate estimate of
. □
Proof of Theorem 1. Firstly, from Lemma 5, we know that as . So by the non-vanishing result of , we obtain the irrationality of .
By Lemmas 5 and 6, we have that for any positive number
,
for
large enough. So Lemma 1 implies that
□
8. Rational Values and Differential Systems
In this section, we discuss the values of the function f for special choices of . In particular, we show that the theorem for fails. We further discuss the differential systems for general .
Let
. Then, the corresponding function
So it takes a rational value for
. On the other hand, this value is not the root of
in Proposition 5. More generally, we obtain the following formula for
when
where
is the Taylor expansion of
up to the term
. So
is rational when
. So in general, the irrationality result fails for general nonzero rational
x when
. Next we will see that this is not the singular points of the differential system appearing in Siegel–Shidlovskii’s theorem.
It is well known that
satisfies a second-order differential equation
For example, see [
18] (Equation (
2), Chapter 6.1). Then, the singular point of this equation is
. For more rational values of confluent hypergeometric functions, see [
19] (Proposition 4.1).
In general, the differential system for
can be obtained as follows. We use notation
for the dependence of
f on parameters
, and
by convention. Then, we have
By induction, we find that
satisfies a linear ordinary differential equation of order
. The only possible singular point is
.
9. Related Results
In conclusion, we prove the irrationality of values for the special kind of generalized hypergeometric functions (see Theorem 1):
if
is not a root of a rational polynomial depending on
; estimates of its irrationality measure can be found as follows:
The mod-
p method could be interesting to generalize to a
p-adic version and obtain more information about rational approximations of
.
After the paper was finished, we learned of a recent preprint by Fischler–Rivoal [
20]. They claimed they had solved a long-standing problem in the theory of Type-
E functions, which states that if a Type-
E function has an irrational value at a rational point, then the irrationality measure of the value is 2. So their result combined with the irrationality part of our main theorem actually implies that the irrationality measure of those values of
is exactly 2, which greatly improves the irrationality measure estimate part of our main theorem.