On two conjectures of Sun concerning Ap\'ery-like series

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Ap\'ery-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few highly non-trivial functional equations of the polylogarithms of weight $3$ and $4$.


Introduction
Let ζ(s) := ∞ n=1 n −s be the Riemann zeta function for Re s > 1.In the 1979's proof [1] of the irrationality of ζ(3), R. Apéry made use of the following infinite series involving central binomial coefficients: Since then, the Apéry-like series have attracted much attention.We refer the reader to [12] for a survey on recent progress.The aim of this paper is to prove two conjectures of Z.-W.Sun concerning Apéry-like series.These conjectures were published first in [10] and included in Sun's book [11].Define the classical harmonic numbers There are two major steps in our proof of this conjecture.First, we will express the Apérytype series on the left-hand side of (1.1) (resp.(1.2)) by some log-sine-cosine (resp.log-sinhcosh) integrals.Then, we will evaluate these integrals using a single-valued version of the polylogarithms, denoted by D m (x) in Zagier's seminal paper [15].
The proof is now complete.
Lemma 2.2.For any positive integer p and real number z ∈ [0, 1/2], we have ) Similarly, for any positive integer p and real number z ∈ [0, 1/2], we have ) Proof.We first prove (2.4).By [6, Eqn.(D.8), pp.52-53], we have where Summing up the two equations, substituting z 2 for z and then multiplying by z, we obtain By the change of variables z = 1 2 sin θ for z ∈ [0, 1/2) and θ ∈ [0, π/2), we arrive at Using (2.1), we obtain (2.9) We observe that Let z = 1 2 sinh θ for z ∈ [0, 1/2) and θ ∈ [0, log( √ 2 + 1)), the above equation can be written as Therefore, For ease of reading, we now outline our proof as the calculations are somewhat involved.We first express the functions Lsc j,k (θ) (j + k = 5) for j < k in terms of the ones with j > k and then show that we can rewrite each of the latter, after subtracting a suitable linear term in θ, in terms of the single-valued function D 4 (Lemmas 3.3 and 3.4).Substituting θ = π 6 and = 5π to match the one on the RHS of (1.1).Moreover, upon realizing that β(4) can be written as D 4 (i), Conjecture (1.1) is reduced to showing the vanishing of a rational linear combination of only D 4 -terms as in (3.27).It then remains to find-and in fact to concoct-suitable functional equations for D 4 which, after an appropriate specialization, match precisely this combination.
Step 1.It is clear from the definition that The special values of Lsc j,k at π have been determined by L. Lewin [8] and [9, Section 7.9].As observed in [3], Lewin's result can be stated in the form In particular, it is known that Step 2. We introduce two different versions of the Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithm [13,14,15]: for |x| ≤ 1, x = 0, 1, where R m = Im for m even and R m = Re for m odd, and where we adopt Zagier's ad hoc convention (p.413 in loc.cit. We extend D m (x) to C \ {0, 1} as a single-valued and real analytic function by the inversion relation (3.9) and we can check that D m (x) satisfies the complex conjugate relation (3.10) below In particular, complex conjugate relation implies that It also satisfies distribution relations as follows: for any positive integer N we have Indeed, this follows easily from the fact that for all |x| ≤ 1 and 1 ≤ j ≤ m we have The following computational lemma will be used repeatedly below.
Lemma 3.1.Let 0 < θ < π.Let f (x) be a rational function of x with real coefficients.Set For any positive integer m let σ m = 2 i , δ m = 0 if m is even and Proof.By definition, we may rewrite D m f (e i θ ) as Thus we have Moving the j = 0 term in the first sum to the end, setting j → j + 1 in the second sum, and combining like terms, we then arrive at The expression for D m in the lemma now follows easily from the definition (3.6).
Turning to D m , we only need to handle the extra term at the end of (3.7).Noticing that Now we can complete the proof of the lemma immediately.
Corollary 3.2.Notation as above.Put For any positive integer m let a ± m (θ) = 1 if m is even, and Proof.By simple calculations, Hence These quickly lead to the equalities in the corollary.
Step 3. Next, we express both Ls 4 and Lsc 3,2 in terms of polylogarithms.
Lemma 3.4.The following expression for Lsc 3,2 (θ) holds for all θ ∈ (0, π): (3.16) Proof.The proof of this lemma is completely similar to that of Lemma 3.3.As above, let A = A(θ) and B = B(θ).By straightforward computations using Corollary 3.2 we find that (which is used to compute the limit as θ → 0), Thus by taking θ → 0 we see that the difference between the left-hand and right-hand sides of by the identities (see [9, (1.16), (6.5) and (6.12)]) Thus we only need to show the second derivatives of both sides of (3.16) agree: (3.17) Now we have LHS of (3.17 by (3.15) and then the distribution relation.This completes the proof of the lemma.
Step 4. We will need the following functional equation of D 4 , which is a variant of Kummer's Li 4 equation [9, Eqn.(7.78)].(Note Λ 4 (x) therein is closely related to Li 4 (−x), in particular it only differs by products of lower weight terms.In order to convert from [9, Eqn.(7.78)] to the D 4 functional equation we essentially only need to add a negative sign to all the arguments from [9, Eqn.(7.78)], and drop any product terms.) Let Then we have F (x, y) := H(x, y) + H(y, x) is mapped to 0 under D 4 .
Proof.In order to verify that D 4 F (x, y) = 0 for all x, y, we apply [15, Proposition 1], which states that if {n i , x i (t)} is a collection of integers n i and rational functions of one variable x i (t), satisfying In this tensor condition the tensors are multiplicative (ab) ⊗ c = a ⊗ c + b ⊗ c, and we can ignore torsion (multiplication by roots of unity) in each slot.This tensor condition is closely related to the ⊗ m -invariant ("symbol") of multiple polylogarithms [7], and amounts to a convenient reformulation of the derivative of D m (x i (t)) for the purposes of calculation.
Set m = 4, and fix y = y 0 ∈ C, it is then straightforward (if tedious) to check that (3.18) vanishes for the list of coefficients and arguments in F (x, y 0 ).Hence for any fixed y = y 0 the combination D 4 (F (x, y 0 )) is constant.By the symmetry of F (x, y) with respect to x ↔ y, we also have by the same calculation that for any fixed x = x 0 , the combination D 4 (F (x 0 , y)) is constant.It follows that D 4 (F (x 1 , y 1 )) = D 4 (F (x 2 , y 1 )) = D 4 (F (x 2 , y 2 )) for any (x 1 , y 1 ), (x 2 , y 2 ) ∈ C 2 , so D 4 (F (x, y)) is constant overall.Since D 4 vanishes on the real line, and by specializing for example x = y = 1 2 all arguments in F (x, y) are real, this constant is necessarily 0. We have therefore established the required functional equation.
Step 5. Specialization to the 12-th roots of unity.In the rest of this section, we put ρ := e 2π i /12 .Note that by applying (3.9) and (3.10) at most twice, we can make the argument of D 4 lie in the upper half unit disk.We will often apply this rule in our calculations below.
By Lemma 3.3 and Lemma 3.4, we have  Since By the distribution relation By specializing Lemma 3.5 to various choices of x, y we obtain further relations between D 4 .In particular, since D 4 vanishes on both 1  3 F (ρ 2 , ρ) and 1 3 F (ρ 2 , ρ 5 ), we have respectively By adding (3.29) and (3.30), we have Therefore, (3.28) is reduced to By the distribution relations, we have The equations (3.32) and (3.33) establish (3.31).Therefore the proof of (1.1) is complete.
Proof.The proof strategy is exactly the same as for Lemma 3.5; we apply the tensor criterion in (3.18) in the case m = 3.This shows that D 3 (G(x)) is constant.To fix the constant, we specialize to x = 0. We find (simplifying only with inversion at the moment) that This is −5 times the left-hand side of (4.12), hence the left-hand side of (4.12) is equal to exactly 0. The proof of (1.2) is complete.
Remark 4.4.It should be noted that the functional equation in Lemma 4.3 has been concocted to give a simple proof of (4.12) in the previous lines.This functional equation can be broken down into a number of smaller functional equations, with slightly more structured coefficients.Specifically Lemma 4.3 is a combination of the following 4 linearly independent functional equations (irreducible within the selected set of arguments), Each of these can be proven in exactly the same way as Lemma 4.3 itself.In fact, the last one (4.13) is (up to inversion) a re-parameterization of the 3-term [9, Equation (6.10)] functional equation D 3 (x) + D 3 (1 − x) + D 3 (1 − x −1 ) = D 3 (1), with x → x 1−x .Remark 4.5.We originally discovered the proof of (1.2) by expressing (4.1) and (4.2) in terms of colored multiple zeta values by applying Au's mechanism developed in [2].Then (1.2) follows from the computer-aided proof using Au's Mathematica package.For the detailed definition and introduction of colored multiple zeta values, see [16,.

Lemma 4 . 3 .
The following linear combination G(x) vanishes identically under D 3 , where