Explicit identities for infinite families of series involving squared binomial coefficients

It is commonly known that integrals containing log-polylog integrands admit representations in terms of special functions such as the Dirichlet eta and Dirichlet beta functions. We investigate two parameterized families of such integrals and in a particular case demonstrate a connection with the Herglotz function. In the process of the investigation we recover some known Euler sum equalities and discover some new identities.

Parameterized Families of PolyLog Integrals

We present a unified approach to obtain new series representations for various classical constants Among others, we prove that $$\log (2) = \frac{{17}}{{24}} + \sum\limits_{k = 2}^\infty {{{( - 1)}^k}} \frac{{{k^2} + k - 1/2}}{{(k - 1)k(k + 1)(k + 2)}}({H_k} - {H_{{{\left[ {k/2} \right]}}})^2}$$
 $$G = - \frac{1}{2} + 2\sum\limits_{k = 1}^\infty {{{( - 1)}^k}\frac{{k(4{k^2} - 5)}}{{(4{k^2} - 1)(4{k^2} - 9)}}{{(2{H_{2k}} - {H_k})}^2}} $$
 , $$\zeta (3) = \frac{{149}}{{144}} + \frac{1}{8}\sum\limits_{k = 2}^\infty {\frac{{(2k + 1)({k^4} + 2{k^3} + 3{k^2} + 2k - 2)}}{{{{(k - 1)}^2}{k^2}{{(k + 1)}^2}(k + 2)}}{{(2H_k^{(2)} - H_{[k/2]}^{(2)})}^2}} $$
 , where $${H_k} = \sum\nolimits_{j = 1}^k {1/j} $$
 and $$H_k^{(2)} = {\sum\nolimits_{j = 1}^k {1/j} ^2}$$
 denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively, and G is the Catalan constant

New Series Representations for Apéry’s and Other Classical Constants

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New Series Representations for Apéry’s and Other Classical Constants