Kam Cheong Au [‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, Journal of Symbolic Computation 130 (2025), Article no. 102241] discovered a powerful methodology for finding new Wilf–Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of $\pi $ and their duals for Dirichlet L-values. In this note, we explain how a modification of Au’s WZ pairs together with a classical analytic argument yields simpler proofs of these results. We illustrate our method with examples elaborated with assistance of Maple code that we have developed.

Consider the family of automorphic L-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted L-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.

We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$.

We introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$. As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

A class of exotic $_3F_2(1)$-series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these $_3F_2(1)$-series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.

We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel ingredients we are able to gain better control of the arithmetic and produce a record irrationality measure for ζ(4).

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\unicode[STIX]{x1D700}$ is any positive real number and $s$ is large enough in terms of $\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$ that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ‘arithmetic–geometric mean (AGM)’ integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.

We record $\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$ functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for $1/\unicode[STIX]{x1D70B}$ and in the computation of mathematical constants.

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_{0}$, we call the resulting series ‘Ramanujan series with the shift $x_{0}$’. Then we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell =1,2,3$ and the shift $x_{0}=1/2$.

We give a sufficient condition for the hypergeometric function $_{3}F_{2}$ to be a linear combination of the logarithm of algebraic functions.

For a certain class of hypergeometric functions $_{3}F_{2}$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.

Some congruences on conjectures of van Hamme are established. These results give partial answers to some of Swisher's conjectures.

We prove new integral representations of the approximation forms in zetavalues.

We establish some supercongruences for the truncated $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series involving the $p$ -adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated $_{3}F_{2}$ hypergeometric series. Related supercongruences modulo $p^{3}$ are proposed as conjectures.

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$ -adic hypergeometric function for any odd prime $d$ .

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.