Related papers: The Partial Theta Operator and Multivariate Genera…
We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two specific primary cases of the…
In a recent preprint, arXiv:1606.05495v1, Alexandrov, Banerjee, Manschot and Pioline introduced generalized error functions and used them to construct indefinite theta series associated to quadratic lattices L of signature (n-2,2). These…
We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series…
We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices, which we call the Tesler function.…
We introduce the family of Theta operators $\Theta_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson for $\Delta_{e_{n-k-1}}'e_n$. We show that…
This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several…
We introduce generalized multipliers for left-invertible operators which formal Laurent series $U_x(z)=\sum_{n=1}^\infty(P_ET^{n}x) \frac{1}{z^n}+\sum_{n=0}^\infty(P_E{T^{\prime*}}^{n}x)z^n$ actually represent analytic functions on an…
We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…
In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized Fox--Wright function and the generalized M-series and…
Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function…
The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. For $q\in (-1,0)$ the…
We introduce a more general discrete fractional operator, given by convex linear combination of the delta and nabla fractional sums. Fundamental properties of the new fractional operator are proved. As particular cases, results on delta and…
In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such…
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 Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…
Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) =…
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.
The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new…
In this article, we prove two identities of generalized Lambert series. By introducing what we call $\mathcal{S}$-series, we establish relationships between multiple generalized Lambert series and multiple infinite products. Compared with…
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these…