Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the $_6ϕ_5$ sum, Ramanujan's $_1ψ_1$ sum and Bailey's $_6ψ_6$ sum.
We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial proof of Watson's identity which implies the Rogers-Ramanujan identities.
By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/π^m$ and $π^m$.
According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $π$-formula, is offered. Furthermore, $q$-analogues of several other $π$-formulas are also established in terms of the $q$-series method and the telescoping method.
By telescoping method, Sun gave some hypergeometric series whose sums are related to $π$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper summability of $r(k)t_k$ for $r(k)$ being a rational function of $k$. We give an upper bound and a lower bound on the degree of the numerator of $r(k)$ such that $r(k)t_k$ is Gosper summable. We also show that the denominator of the $r(k)$ can read from the Gosper representation of $t_{k+1}/t_k$. Based on these results, we give a systematic method to construct series…
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a naïve version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.
We prove that the set of permutations sorted by a stack of depth $t \geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infinite.
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. …
We prove that the set of permutations generated by a stack of depth two and an infinite stack in series has a basis (defining set of forbidden patterns) consisting of 20 permutations of length 5, 6, 7 and 8. We prove this via a ``canonical'' generating algorithm.
We prove that the class of permutations generated by passing an ordered sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free language to compute the generating function: \begin{align*} \sum_{n\geq 0} c_n t^n &= \frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q} \end{al…
Let $β=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(δ(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $δ(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}ε_jβ^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}ε_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(δ(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fib…
We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.
Since the study by Jacobi and Hecke, Hecke-type series have received extensive attention. Especially, Hecke-type series involving infinite products have attracted broad interest among many mathematicians including Kac, Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain the truncated versions of these series as well as some other known series of the same type. As consequences, three families of inequalities for certain p…
In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z) = \sum_n a_nz^n$ is an infinite series with $a_n \geq 1$ and $a_0 + \cdots + a_n = O(n + 1)$ for all $n$, we prove that a super-polynomially long initial segment of $f^k(z)$ is log-concave. Furthermore, if there exists constants $C > 1$ and $α< 1$ such that $a_0 + \cdots + a_n = C(n + 1) - R_n$ where $0 \leq R_n \leq O((n + 1)^α)$, we show that an exponentially long initial segment of $f^k(z)$ is log-concave. This resolves a conjecture proposed by Letong Hong and the author, w…
A Delaunay polytope $P$ is said to be {\em extreme} if the only (up to isometries) affine bijective transformations $f$ of $\R^n$, for which $f(P)$ is again a Delaunay polytope, are the homotheties. This notion was introduced in \cite{DGL92}; also some examples in dimension 1, 6, 7, 15, 16, 22, 23 were constructed and it was proved that in dimension less than 6 there are no extreme Delaunay polytopes, except the segment. In this note, for every $n\geq 6$ we build an extreme Delaunay polytope $ED_n$ of dimension $n$.
Lueck expressed the Gromov norm of a knot complement in terms of an infinite series that can be computed from a presentation of the fundamental group of the knot complement. In this note we show that Lueck's formula, applied to torus knots, yields surprising power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of 4.
In this paper, we investigate several infinite products with vanishing Taylor coefficients in arihmetic progressions. These infinite products are closely related to the Rogers--Ramanujan continued fraction. Moreover, a handful of new identities involving Ramanujan's parameters will be established.
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $π$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.
