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

Abstract

In this paper, we study Apéry-type series involving the central binomial coefficientsand its variations where the summation indices may have mixed parities and some or all ">" are replaced by "≥", as long as the series are defined. We show that all these sums can be expressed as Q-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. We also show that the corresponding series where 2n 1 n 1 /4 n is replaced by 2n 1 n 1 2 /16 n can be expressed in a similar way except for an extra factor of 2/π.

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

“…It is natural to see if the same idea works for the binomial series, too. In [15] we successfully carried out this investigation and proved that results similar to those in Thm. 1.2 and its odd-indexed versions, Thm.…”

“…The above iteration formulas form the foundation of all the results in [15]. Set a − m (x) = (−1) m a m (x).…”

“…By repeatedly applying (4.2)-(4.5) we can find many results analogous to those in section 2. See [15] for the approach we used to study the non-alternating version. Fix a primitive 8th root of unity µ = (1 + i)/ √ 2.…”

“…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.…”

On a problem involving the squares of odd harmonic numbers

“…To show this, we use an interated integral-based approach, using results from [23]. We recall that for any real numbers a, b and 1-forms f j (t) dt (j = 1, .…”