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 PirastuStrehl [15] and Abramov [1, 2] gave the minimal decomposition when is a rational function, AbramovPetkovšek [3, 4] gave the minimal decomposition when is a hypergeometric term, and ChenHuangKauersLi [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.
We will see that the degeneration is closely related to the degrees of the elements in .
Lemma 2.2
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 subring of in general. Let denote the coset of a polynomial . We see that the quotient space is finite dimensional.
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 nonnegative 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 .
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 nonnegative 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
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