"Divergent"Ramanujan-type supercongruences

"Divergent"Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique.

Here · p is the Legendre symbol and the Pochhammer notation (a) b is used for denoting Γ(a + b)/Γ(b) also in the cases when b is not a non-negative integer; of course, if b = n ∈ Z ≥0 we have, as usual, (a) n = n−1 k=0 (a+k) with the convention that the empty product equals 1. The question mark indicates that the corresponding supercongruence remains conjectural; the non-questioned entries (1)-(3) are proved in this paper by extending the method of [19], while the supercongruence (4) (even in a more general form) is shown by Zhi-Wei Sun in his preprint [14].
Note that we can sum in (1), (2), (4), (3), and (8) up to p−1 2 , since the p-adic order of ( 1 2 ) n /n! is 1 for n = p+1 2 , . . . , p − 1. Main theorem. The following supercongruences take place: We find quite illogical that our strategy based on the creative Wilf-Zeilberger theory [11] of WZ-pairs allows us to do only three entries from the list (1)-(9); a very similar lack of luck was reported in [19]. Although we have WZ-pairs for (4)- (8) as well, they seem to be quite helpless for showing the corresponding congruences modulo the expected powers of p. Because of the clear relationship of such congruences with Ramanujan's formulae for 1/π and their generalizations (see [19] and Section 4), we do expect a more universal method for proving the Ramanujan-type supercongruences.
In Section 2 we present auxiliary congruences, some of them are remarkable in their own. Section 3 contains the proofs of (10)- (12). The final Section 4 reviews the "divergent" Ramanujan-type series for 1/π and 1/π 2 as our motivation to the the above list of supercongruences.
After posting the preprint online we were informed by Zhi-Wei Sun that he had experimentally and independently discovered the congruences (1)- (8), but also proved in [14] a more general than in (4) supercongruence. We thank him for attracting our attention to his work.

Precongruences
In this section we summarize our needs for proving the supercongruences of the main theorem. Lemma 1. The following congruences are valid : Proof. The first congruence follows from specialization N = (p − 1)/2 of Staver's identity [13] N n=1 2n n The second congruence is the modulo p reduction of Tauraso's congruence in [16,Theorem 4.2]. It is interesting to mention that the latter follows from the N = p specialization of another combinatorial identity, conjectured by Borwein and Bradley [3] and proved by Almkvist and Granville [1].
For a prime p > 3, let x be a rational number such that both x and 1 − x do not involve p in their prime factorizations and 1 − x is a quadratic residue modulo p. Take y such that Proof. It is well known that and, by replacing x with −x and taking the appropriate linear combination of the two expressions, Consider
Proof. It is shown in [15, Theorem 1.2] that for m in Z * p , Although the theorem is stated for m ∈ Z only, the proof does not make use of this integrality: we can apply it for m = 1/2 as well. In this case V k (1/2) = 1 + (−1) k /2 k , so that the right-hand side of (32) becomes (30) if we additionally use 2 p−1 ≡ 1 (mod p).
The congruence (31) is clear for p = 3, while for p > 3 it follows from specialization x = −8, y = 3 of (19) and noting that

Proofs of the supercongruences
Proof of (10). Take and (23), so that F (n, k) and G(n, k) form a WZ-pair. Summing (23) over n = 0, 1, . . . , p−1 2 , we obtain Furthermore, for k = 1, 2, . . . , p−1 2 we have because each of the three Pochhammer products in the numerator is divisible by p while the denominator, p−1 2 ! 3 , is coprime with p. Comparing this result with (34), as in the proof of Theorem 1 in [19], we see that hence we can replace, modulo p 3 , our sum (33) by (Note that, in contrast with the proofs in [19], the newer sum is not reduced to a single term.) Comparing the resulted expression (35) for (33) we see that (10) is equivalent to On noting that (1 + p 2 ) n−1 ≡ (1) n−1 (mod p), we reduce (36) to its equivalent which is exactly (13).

"Divergent" Ramanujan-type series
In [19], the second-named author generalized an observation of L. Van Hamme about Ramanujan-type identities for 1/π and 1/π 2 . The idea is to associate with each such identity where a, b, c, z, and r are rational and A n is a related Pochhammer ratio (or, more generally, an Apéry-like sequence; cf. [19]), the supercongruence respectively, for all p ≥ p 0 . Recently [9], the first-named author went even further and considerably extended the pattern; however this remains an unproven observation. The general machinery for proving Ramanujan-like series for 1/π [2,4,18] produces, in several cases, divergent instances like The summations in (43) have to be understood as the analytic continuation of the corresponding 3 F 2 -hypergeometric series; for example, the second formula in (43) can be given by 1 2πi  The first appearance of divergent series for 1/π is [2, p. 371]. In view of the observation from [19], the formulae in (43) motivate our supercongruences (1) and (3), respectively. and a = 1/4; the values τ and k are found from (46) and c, b and a from (47). The resulting set corresponds to the series ∞ n=0 ( 1 2 ) 5 n (1) 5 n (10n 2 + 6n + 1)(−1) n 2 2n "=" 4 π 2 with the left-hand side understood as the analytic continuation of the participating hypergeometric series to C \ [1, +∞). A similar duality for the 3 F 2 -evaluations of 1/π can be explained by the modular origin of the corresponding hypergeometric series, like the one we give for (43). The duality mechanism for the 5 F 4 -examples remains a mystery.
As already pointed out in [19], all Ramanujan-type series for 1/π and their generalizations possess unexpectedly strong arithmetic properties. In particular, these are reflected by the supercongruences for truncated sums -it is probably not surprising to see the examples (1)- (9). What is more remarkable, the p-analogues make no difference of their origin: whether they come from convergent or divergent formulae. This kind of democracy as well as an apparent simplicity of the supercongruences make them an attractive object for further investigation.