Related papers: Finite Form of the Quintuple Product Identity
The quintuple product identity are deduced from a new semi-finite form, which are obtained from the very-well-poised $_6\phi_5$ series.
In this paper, we present the bilateral truncated identity of the quintuple product identity, which is a generalization of the truncated quintuple product identities given by Chan, Ho and Mao [J. Number Theory 169 (2016) 420--438].…
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula.…
A surprising simple result about quadrilaterals is given as an application of the vector triple product identity.
Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product…
By considering a limiting form of the q-Dixon_4\phi_3 summation, we prove a weighted partition theorem involving odd parts differing by >= 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of…
The q-binomial formula in the limit q->1 is shown to be equivalent to the Rogers five term dilogarithm identity.
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple…
In this article the infinite product of bicomplex numbers is defined and the convergence and divergence of this product are discussed.
Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite…
Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's…
Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer…
We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.
We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups…
We give generalizations of a finite version of Euler's pentagonal number theorem and of a q-identity of Gauss.
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 paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for…
Jacobi's triple product identity is proved from one of Euler's $q$-exponential functions in an elementary way.
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both…
It is shown that the idempotent completion of the additive hull of the tensor product of the residue category of the category of paths of a locally finite quiver modulo an admissible ideal and a dualizing category is dualizing. Furthermore,…