Polynomial Reduction and Super Congruences

07/17/2019
by   Qing-Hu Hou, et al.
Tianjin University
0

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. As applications, we derived two infinite families of super-congruences.

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1 Introduction

In recent years, many super congruences involving combinatorial sequences are discovered, see for example, Sun [16]. The standard methods for proving these congruences include combinatorial identities [18], Gauss sums [5], symbolic computation [14] et al.

We are interested in the following super congruence conjectured by van Hamme [19]

where is a odd prime and is the rising factorial. This congruence was proved by Mortenson [13] Zudilin [21] and Long [12] by different methods. Sun [17] proved a stronger version for prime

where is the -th Euler number defined by

A similar congruence was given by van Hamme [19] for :

Long [12] showed that in fact the above congruence holds for arbitrary odd prime modulo . Motivated by these two congruences, Guo [8] proposed the following conjectures (corrected version).

Conjecture 1.1
  • For any odd prime , positive integer and odd integer , there exists an integer such that

    (1.1)
  • For any odd prime , positive integer and odd integer , there exists an integer such that

    (1.2)

Liu [11] and Wang [20] confirmed the conjectures for and some initial values . Jana and Kalita [10] and Guo [9] confirmed (1.1) for and . We will prove a stronger version of (1.1) for the case of and arbitrary odd and a weaker version of (1.2) for the case of and arbitrary odd by a reduction process.

Recall that a hypergeometric term is a function of such that is a rational function of . Our basic idea is to rewrite the product of a polynomial in and a hypergeometric term as

where are polynomials in such that the degree of is bounded. To this aim, we construct such that equals the product of and a polynomial and that and has the same leading term. Then we have

is the product of and a polynomial of degree less than . We call such a reduction process one reduction step. Continuing this reduction process, we finally obtain a polynomial with bounded degree. We will show that for , and an arbitrary polynomial of form with odd, the reduced polynomial can be taken as . This enables us to reduce the congruences (1.1) and (1.2) to the special case of , which is known for .

We notice that Pirastu-Strehl [15] and Abramov [1, 2] gave the minimal decomposition when is a rational function, Abramov-Petkovšek [3, 4] gave the minimal decomposition when is a hypergeometric term, and Chen-Huang-Kauers-Li [6] applied the reduction to give an efficient creative telescoping algorithm. These algorithms concern a general hypergeometric term. While we focus on a kind of special hypergeometric term so that the reduced part has a nice form.

The paper is organized as follows. In Section 2, we consider the reduction process for a general hypergeometric term . Then in Section 3 we consider those with the property is a shift of , where . As an application, we prove a stronger version of (1.1) for the case . Finally, we consider the case of is a shift of , which corresponds to (1.2). In this case, we show that there is a rational number instead of an integer such that (1.2) holds when .

2 The Difference Space and Polynomial Reduction

Let be a field and be the ring of polynomials in with coefficients in . Let be a hypergeometric term. Suppose that

where . It is straightforward to verify that

(2.1)

We thus define the difference space corresponding to and to be

We see that for , we have for a certain polynomial .

Let denote the set of nonnegative integers and the set of integers, respectively. Given , we denote

(2.2)
(2.3)

and

(2.4)

where denotes the leading coefficient of .

We first introduce the concept of degeneration.

Definition 2.1

Let and be given by (2.2) and (2.4). If

we say that the pair is degenerated.

We will see that the degeneration is closely related to the degrees of the elements in .

Lemma 2.2

Let and be given by (2.3) and (2.4). For any polynomial , let

If is degenerated and , then ; Otherwise, .

Proof. Notice that

If the leading terms of and do not cancel, the degree of is . Otherwise, we have and

i.e., .  

It is clear that is a subspace of , but is not a sub-ring of in general. Let denote the coset of a polynomial . We see that the quotient space is finite dimensional.

Theorem 2.3

Let and be given by (2.3) and (2.4). We have

Proof. For any nonnegative integer , let

We first consider the case when the pair is not degenerated. By Lemma 2.2, we have

Suppose that is a polynomial of degree . Then

(2.5)

is a polynomial of degree less than and . By induction on , we derive that for any polynomial of degree , there exists a polynomial of degree such that . Therefore,

Now assume that is degenerated. By Lemma 2.2,

The above reduction process (2.5) works well except for the polynomials of degree . But in this case,

is a polynomial of degree less than . Then the reduction process continues until the degree is less than . We thus derive that

completing the proof.  

Example 2.1

Let be a positive integer and

where is the raising factorial. Then

and

We have

is of dimension one.

3 The case when

In this section, we consider the case when and has a symmetric property. We will show that in this case, the reduction process maintains the symmetric property. Notice that in this case

has the same degree as , the pair is not degenerated.

We first consider the relation between the symmetric property and the expansion of a polynomial.

Lemma 3.1

Let and . Then the following two statements are equivalent.

  • (, respectively).

  • is the linear combination of (, respectively).

Proof. Suppose that

Then

Therefore,

The case of can be proved in a similar way.  

Now we are ready to state the main theorem.

Theorem 3.2

Let such that

for some . Then for any non-negative integer , we have

and

where

(3.1)

Proof. We only prove the case of . The case of can be proved in a similar way. By Lemma 3.1, we may assume that

where is even and are the coefficients.

Since is not degenerated, taking

(3.2)

in Lemma 2.2, we derive that

(3.3)

is a polynomial of degree . More explicitly, we have

is a polynomial with leading term .

Notice that

and

Since , i.e., , we deduce that

By the relation (3.1), we derive that

Suppose that is a linear combination of the even powers of and . By Lemma 3.1, we have and thus

also satisfies since and are both even. It is clear that and the degree of is less than the degree of . Continuing this reduction process, we finally derive that for some polynomial with degree and satisfying . Therefore,

Suppose that is a linear combination of the odd powers of and . Then we have and thus

also satisfies . Continuing this reduction process, we finally derive that

This completes the proof.  

We may further require to express as an integral linear combination of when .

Theorem 3.3

Let

where is a positive integer and is a rational number with denominator . Then for any positive integer , there exist integers and a polynomial such that

Moreover, if .

Proof. We have

Let

We see that it is the case of and of Theorem 3.2. From (2.1), we derive that

(3.4)

where and are given by (3.2) and (3.3) respectively. Multiplying on both sides, we obtain

(3.5)

where ,

(3.6)

and

(3.7)

Notice that and is a monic polynomial of degree . Moreover, contains only even powers of or only odd powers of . Using to do the reduction (2.5), we derive that there exists integers such that

becomes a polynomial of degree less than . Clearly, . Replacing by and multiplying , we derive that

completing the proof.  

As an application, we confirm Conjecture 6 of [11].

Theorem 3.4

Let

For any positive odd integer , there exist integers and such that

holds for any prime .

Proof. Taking and in Theorem 3.3, there exist an integer and a polynomial such that

where . Summing over from to , we derive that

where . Noting that

and

we have

Hence

Let . We then have

Sun [17] proved that for any prime ,

Therefore,

Remark 1. The coefficient and the polynomial can be computed by the extended Zeilberger’s algorithm [7].

4 The case when

We first give a criterion on the degeneration of .

Lemma 4.1

Let such that . Suppose that . Then is not degenerated.

Proof. Let and

It is clear that the coefficient of in is and the coefficient of in is . Since , we derive that . Thus,

Since , the pair is not degenerated.  

When is a shift of , we have a result similar to Theorem 3.2.

Theorem 4.2

Let such that

for some . Assume further that . Then for any non-negative integer , we have

and

where

Proof. The proof is parallel to the proof of Theorem 3.2. Instead of (3.2), we take

in Lemma 2.2. By Lemma 4.1, is not degenerated and

Hence the polynomial

satisfies

Moreover, we have

so that the reduction process maintains the symmetric property. Therefore, the reduction process continues until the degree is less than .  

Similar to Theorem 3.3, we have the following result.

Theorem 4.3

Let

where is a positive integer and is a rational number with denominator . Suppose that . Then for any positive integer , there exist integers and a polynomial such that

where

Moreover, if .

Proof. The proof is parallel to the proof of Theorem 3.3. Instead of (3.6) and (3.7), we take

(4.1)

and

(4.2)

so that (3.5) still holds. It is clear that . But in this case, is not monic. The leading term of is

Now let us consider the reduction process. Let be a polynomial of degree . Assume further that contains only even powers of or only odd powers of . Setting

we see that and . Since contains only even powers of or only odd powers of , so does . Therefore, .

Continuing this reduction process until , we finally obtain that there exist integers such that

is a polynomial of degree less than and with integral coefficients, where is the product of the leading coefficient of

as desired.  

For the special case of , we may further reduce the factor .

Lemma 4.4

Let be a positive integer and

  • If is odd, then there exist an integer and a polynomial such that

    where .

  • If is even, then there exist integers and a polynomial