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Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)}…
In this appendix to our paper with the same title posted on arxiv we give a quick proof of an inequality that can be substituted to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but also appears…
The R\'enyi (Shannon) entropy, i.e. $Re_{\alpha}(Sh)$, of the ground state of quantum systems in local bases normally show a volume-law behavior. For a subsystem of quantum chains at critical point there is an extra logarithmic subleading…
We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the…
We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the…
We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum_{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate…
The basic aspects of both Boltzmann-Gibbs (BG) and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional ($S_{BG} =-k\sum_i p_i \ln p_i$ for the BG formalism) with the…
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…
A $q$-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent $1/(1-q)$ ($q\neq 1$). The limit case $q=1$ recovers a Gaussian…
$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…
We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy $S_q=k\frac{1-\sum_{i}p_i^q}{q-1} (q \in {\cal R}; S_1=-k\sum_{i}p_i \ln p_i)$. Among others, we focus on possible…
We consider the q-hypergeometric equation with q^{N}=1 and $\alpha, \beta, \gamma \in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the…
It was recently published by M. Nauenberg [1] a quite long list of objections about the physical validity for thermal statistics of the theory sometimes referred to in the literature as {\it nonextensive statistical mechanics}. This…
We show that if one uses the invariant form of the Boltzmann-Shannon continuous entropy, it is possible to obtain the generalized Pareto-Tsallis density function, using an appropriate "prior" measure m_{q}(x) and a "Boltzman constraint"…
Let $q, r, s \geqslant 1$ satisfy $\frac{1}{2q} + \frac{1}{2r} = \frac{1}{s}$ and $X \in \mathcal{C}_s(\mathcal{H})$. If $(\lambda_n)_{n=1}^{\infty}, (w_n)_{n=1}^{\infty}$ are sequences in $(0,+\infty)$ and $(\lambda_n^{(1-q)/(2q)}…
We revisit the derivation of a formula for the $q$-generalised multinomial coefficient rooted in the $q$-deformed algebra, a foundational framework in the study of nonextensive statistics. Previous approximate expressions in the literature…
The classic central limit theorem and $\alpha$-stable distributions play a key role in probability theory, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the…
The thermal-equilibrium canonical distribution is currently obtained by maximizing the Boltzmann-Gibbs-von Neumann-Shannon entropy $S_{BG}(p)=k\sum^{W}_{i=1}p_{i}\ln 1/p_{i}$ constrained to $\sum^{W}_{i=1}p_{i}=1$ and…
We use a new $q$-exponential operator based on the $q^{\pm1}$-derivative $\D_{q^{\pm1}}$ of order 1 to derive summation formulas for bilateral basic hypergeometric series ${}_{0}\psi_{1}$, ${}_{1}\psi_{1}$, ${}_{1}\psi_{2}$, and…
There exists only one generalization of the classical Boltzmann-Gibbs-Shannon entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the corresponding extension of the classical…