Related papers: Some truncated identities of Gauss
In this note, we investigate J.-C. Liu's work on truncated Gauss' square exponent theorem and obtain more truncations. We also discuss some possible multiple summation extensions of Liu's results.
In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem, which opened up a new study on truncated theta series. In particular, some truncated versions of a identity of Gauss have been proved. In this…
Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving…
Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting…
Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…
In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
Andrews and Merca introduced and proved a $q$-series expansion for the partial sums of the $q$-series in Euler's pentagonal number theorem. Kolitsch, in 2022, introduced a generalization of the Andrews-Merca identity via a finite sum…
By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number…
Some generalized multi-sum Chu-Vandermonde identities are presented and proved, generalizing some known multi-sum Chu-Vandermonde identities from literature and adding some quadratic and cubic examples of these identities. Some other…
Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of…
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured by Guo and Schlosser and recently confirmed by Jana and…
In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity…
The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP)…
We give generalizations of a finite version of Euler's pentagonal number theorem and of a q-identity of Gauss.
We introduce a kind of finite truncation of the hypergeometric series and provide its discretized integral representation. This is motivated by recent results of Maesaka-Seki-Watanabe and Hirose-Matsusaka-Seki on the identity between…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
In this paper, we mainly establish two supercongruences involving truncated hypergeometric series by using some hypergeometric transformation formulas. The first supercongruence confirms a recent conjecture of the second author. The second…
Recently, Wang and Ma propose a conjecture associated with the possible generalization of Andrews-Warnaar identities. It is confirmed in this paper. As the applications of this conjecture, we prove that a family of series can be expressed…
We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.
Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a…