Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I

Abstract

In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let b n = 4 n / 2n n . Let s 1 , . . . , s d be positive integers with s 1 ≥ 2. We consider the seriesand the variants with some or all indices n j replaced by 2n j ± 1 and some or all ">" replaced by "≥", provided the series are defined. We can also replace b n 1 by its square in the above series when s 1 ≥ 3. The main result is that all such series are Q-linear combinations of the real and/or the imaginary parts of some colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity.

Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, II, , with Two Appendices

“…We first claim that we may reduce the sums in the first two cases to those sums where l j (x) = 2x − 1 appears only when j = 1, if it ever appears. We can prove this by induction on the depth in exactly the same way as was used in the proof of [15,Thm. 4…”

Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values

“…The main results of the present paper can be used to extend some of those concerning the Apeŕy-type series in [28]. See [1,3,4,[28][29][30] and the references therein for recent results on Apeŕy-type series and colored multiple zeta values.…”

Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, III

“…The key idea is to follow the proof of [14,Theorem 4.3] step by step but use the substitution t → i(1 − t 2 )/(1 + t 2 ) (instead of t → (1 − t 2 )/(1 + t 2 ) in the original proof) at the end.…”

“…Moreover, oddindexed variations of both types appeared implicitly, too. See [6, (1.1)] and [14,Remark 4.2] for the former and [8, (A.25)] and [14,Eq. (1.3)] for the latter.…”

“…In our previous work [13,14,15] we have also studied both types and considered their evenodd-indexed variations. Our main result is that many such series can be expressed as Q-linear combinations of the real and/or the imaginary part of colored multiple zeta values of level 4.…”