Related papers: Explicit transformations for generalized Lambert s…

We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including…

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector…

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence…

We introduce a deformation of the Fourier transform on $\mathbb{R}^N$ arising from a representation-theoretic construction associated with $\widetilde{SL}(2,\mathbb{R}) \times O(N)$ that still admits an underlying degree-one operator…

Utilizing inverse Mellin transform of the symmetric square $L$-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function…

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…

We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^\beta = 1 - e^{-xy}$, with $x>0$ and $ \beta > 1$. Solutions to this equation can be found in terms of a certain continued exponential.…

The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mez\"{o} and Baricz, and has found applications in delay differential equations and physics. In this article we…

In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the…

We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan…

Let $\mathcal S^2$ be the Stepanov space and let $ \lambda_n\uparrow\infty$. Let $(a_n)_{n\ge 1}$ be satisfying Wiener's condition $A:= \sum_{n\ge 1} \big(\sum_{k\, :\, n\le \lambda_k \le n+1}|a_k|\big)^2 <\infty$. We prove that $\big\|…

We give the exact distribution of the average of n independent beta random variables weighted by the selected cuts of (0, 1) by the order statistics of a random sample of size n-1 from the uniform distribution U(0,1), for each n. A new…

We develop a new theory of $L$-series based on replacing Dirichlet characters mod $N$ by symmetric functions mod $N$ in order to calculate explicitly the sums of infinite series more closely related to $\zeta(2n+1)$, specifically…

Dirichlet's $L$-functions are natural extensions of the Riemann zeta function. In this paper we first give a brief survey of Ap\'ery-like series for some special values of the zeta function and certain $L$-functions. Then, we establish two…

We have generalized the \textsc{Mathematica} function \texttt{Apart} from 1 to $N$ dimension, the generalized function \texttt{\$Apart} can decompose any linear dependent elements in $\mathcal{V}_{x}^*$ to irreducible ones. The elements in…

Given a formal power series f(z) we define, for any positive integer r, its rth Witt transform, W_f^{(r)}, by rW_f^{(r)}(z)=sum_{d|r}mu(d)f(z^d)^{r/d}, where mu is the Moebius function. The Witt transform generalizes the necklace…

We interpret the "explicit formula" in the sense of analytic number theory for the zeta function of an ordinary abelian variety of dimension g over a finite field as a transversal index theorem on a (2g+1)-dimensional Riemannian foliated…

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote…

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special…