Related papers: An Inversion Formula for Multivariate Power Series

Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.

Let $f(x,y)$ be a complex irreducible formal power series without constant term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and…

We consider the $n\times n$ Hankel matrix $H$ whose entries are defined by $H_{ij}=1/s_{i+j}$ where $s_k=(k-1)!$ and prove that $H$ is invertible for all $n\in\mathbb{N}$ by providing an explicit formula for its inverse matrix.

We present explicit formulas for Moore-Penrose inverses of some families of set inclusion matrices arising from sets, vector spaces, and designs.

We obtained the formulas for the quantities of positive, negative and zero values of the Mobius function for any real x in terms of the Mobius function values for square root of x - similar to the identities we found earlier for the Mertens…

Finding a computationally efficient algorithm for the inverse continuous wavelet transform is a fundamental topic in applications. In this paper, we show the convergence of the inverse wavelet transform.

This short note develops fundamental functions associated with the scattered shifts of the inverse \emph{multiquadric} function $(\alpha^2 + x^2)^{-k}$, for $k\in\mathbb{N}$.

While a characterization of unavoidable formulas (without reversal) is well-known, little is known about the avoidability of formulas with reversal in general. In this article, we characterize the unavoidable formulas with reversal that…

Let $k, l, a$ and $b$ be positive integers with $\max\{a, \, b\}\ge2$. In this paper, we show that every positive rational number can be written as the form $\varphi(km^{a})/\varphi(ln^{b})$, where $m, \, n\in\mathbb{N}$ if and only if…

We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx. $$ This formula is motivated by the results of Fairlie…

In this paper we investigate the series $\sum_{k = 1}^\infty \binom{3k}{k}^{-1}k^{-n}x^k$. Obtaining some integral representations of them, we evaluated the sum of them explicitly for $n = 0, 1, 2$.

In this paper we present a method for constructing multiple-sum $q$-series for what is known as Mixed Mock Modular forms. We also present some multi-sum analogues of the Durfee identity, and discuss a construction of its combinatorial…

We consider the inverse problem of the broken ray transform (sometimes also referred to as the V-line transform). Explicit image reconstruction formulas are derived and tested numerically. The obtained formulas are generalizations of the…

The inversion in the sphere or Kelvin transformation, which exchanges the radial coordinate for its inverse, is used as a guide to relate distinct electrostatic problems with dual features. The exact solution of some nontrivial problems are…

We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$,…

Using combinatorial techniques, we derive a recurrence identity that expresses an exponential power sum with negative powers in terms of another exponential power sum with positive powers. Consequently, we derive a formula for the power sum…

In this paper will be considered standard forms of generalized inverses for matrices in the shape of block representations {1, 2, 3, 4, 5, 5^k}-inverse. Especially will be considered Moore-Penrose inverse and the group inverse. Results from…

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

This article primarily aims to unify the various formalisms of multivariate coefficients of variation, leveraging advanced concepts of generalized means, whether weighted or not, applied to the eigenvalues of covariance matrices. We…

In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the generalized Euler function $ \varphi_e(n) $, which is determined by some…