Related papers: Powers of Euler's q-Series

For noncommutative variables x,y an expansion of log(exp(x)exp(y)) in powers of x+y is obtained.Each term of the series is given by an infinite sum in powers of x-y.The series is represented by diagrams.

In this paper, a new exponential and logarithm related to the non-extensive statistical physics is proposed by using the q-sum and q-product which satisfy the distributivity. And we discuss the q-mapping from an ordinary probability to…

The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval $\Dt$ was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as…

Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed…

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results…

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

Let x be a CM point on a modular or Shimura curve and p a prime of good reduction, split in the CM field K. We define an expansion of an holomorphic modular form f in the p-adic neighborhood of x and show that the expansion coefficients…

For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double…

The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of…

For two real bases $q_0, q_1 > 1$, we consider expansions of real numbers of the form $\sum_{k=1}^{\infty} i_k/(q_{i_1}q_{i_2}\cdots q_{i_k})$ with $i_k \in \{0,1\}$, which we call $(q_0,q_1)$-expansions. A sequence $(i_k)$ is called a…

We present an asymptotic formula for the number of line segments connecting q+1 points of an nxn square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly…

Let $q\in(1,2)$; it is known that each $x\in[0,1/(q-1)]$ has an expansion of the form $x=\sum_{n=1}^\infty a_nq^{-n}$ with $a_n\in\{0,1\}$. It was shown in \cite{EJK} that if $q<(\sqrt5+1)/2$, then each $x\in(0,1/(q-1))$ has a continuum of…

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree $n$ and parameters $\alpha$ and $\beta$. From these new results, asymptotic expansions of the zeros are derived and methods are…

In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$.

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…