Volume 89, Nº 6 (2025)

For polynomials $Q_{n}(z):=z^n + \cdots$ defined by three-term recurrence relations $Q_{n+1}=zQ_n-a_{n-p+1}Q_{n-p}$, $p\ge {1}$, of order $p+1$ with the coefficient $a_{n}\equiv a_{n,N}$ (the variable recurrence coefficient) depending on the parameter $N$, the weak asymptotics of $Q_n (z)$ are investigated in the quasi-classical regime as $n \to \infty$, $n/N \to t$, and $a_{n,N} \to a(t)$. The case $p=1$ (orthogonal polynomials) was studied earlier. The results obtained (for $p=2$) are applied to the problem of eigenvalues distributions of ensembles of normal random matrices.

In this paper, we consider various classes of the abstract fractional difference inclusions with Weyl fractional derivatives and Riemann–Liouville fractional derivatives. We also provide some new results about the well-posedness of abstract integer-order difference inclusions with Euler forward operators, paying a special attention to the analysis of the existence and uniqueness of almost periodic and almost automorphic type solutions to abstract fractional difference inclusions.

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from traditional scenarios is evident here: despite reduced regularity in observables, our method still maintains exponential convergence. In particular, we develop new techniques that yield very precise rates of exponential convergence, as evidenced by numerical simulations. Furthermore, this innovative approach extends to quantitative analyses involving different weighting functions employed by others, surpassing the limitations inherent in prior research. It also enhances the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits via analytic observables. To the best of our knowledge, this is the first result on the uniform exponential acceleration beyond averages along quasi-periodic or almost periodic orbits, particularly from a quantitative perspective.

Recently R. Khan and M. Young proved an order-sharp Lindelöf estimate on the second moment of central values of Maass form symmetric-square $L$-function on the interval $T<|t_j|, where $t_j$ is a spectral parameter of the Maass form. We provide another proof of this result.