By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun [‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7]. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.

In 2011, Guillera [‘A new Ramanujan-like series for $1/\pi ^2$’, Ramanujan J. 26 (2011), 369–374] introduced a remarkable rational ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall’s ${}_{5}F_{4}$-sum. Another proof of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall’s sum together with the WZ method. Subsequently, Chen and Chu introduced a q-analogue of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series. The many past research articles concerning Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational ${}_{7}F_{6}( \frac {27}{64} )$- and ${}_{6}F_{5}( \frac {27}{64} )$-series for $\sqrt {2}$.

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.

In this paper we derive formulas for summation of series involving J. Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H. M. Srivastava. These series are expressed in terms of the Riemann $\zeta$ function and Dirichlet functions $\eta$, $\lambda$, $\beta$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Let B denote the unit ball in ℂn, and dV(z) normalized Lebesgue measure on B. For α > -1, define dVα(z) = (1 - \z\2)αdV(z). Let (B) denote the space of holomorhic functions on B, and for 0 < p < ∞, let p(dVα) denote Lp(dVα) ∩ (B). In this note we characterize p(dVα) as those functions in (B) whose images under the action of a certain set of differential operators lie in Lp(dVα). This is valid for 1 < p < oo. We also show that the Cesàro operator is bounded on p(dVα) for 0 < p < oo. Analogous results are given for the polydisc.