Chuanan Wei, Guozhu Ruan
With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $π$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $π$ in this paper. Similarly, we also establish several $q$-analogues of them.
We establish a q-analogue of Wolstenholme's harmonic series congruence.
The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations
本文通过多参数方法构造$q$-类比，用于加速超几何级数，特别是针对通用常数如$\pi$。
By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalsch{ü}tz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $π$ and $1/π$, including some typical ones discovered by Ramanujan (1914) and Guillera.
On two conjectural series for $π$ and their $q$-analogues
本研究采用算子方法证明了两个涉及二阶调和数的$\pi$猜想级数，并给出了六个$\pi$公式的$q$-模拟结果。
We obtain $q$-analogues of several series for powers of $π$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{π^3}{32}$$ has the following $q$-analogue: \begin{equation*} \sum_{k=0}^\infty(-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty}^2(q^4;q^4)_{\infty}^6} {(q;q^2)_{\infty}^4}, \end{equation*} where $q$ is any complex number with $|q|<1$. We also give $q$-analogues of four new series for powers of $π$ found by the second author.
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$π$ and other eleven nice $π$-formulas. Meanwhile, $q$-analogues of three beautiful series for $ζ(3)$ are also given in the same way.
In this paper, we present a $q$-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the $q$-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given $q$-hypergeometric term. The structure theorem enables us to generalize the $q$-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of $q$-identities. As applications, several $q$-analogues of series for $π$ are presented.
From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.
Let $(W,S)$ be a Coxeter system and write $P_W(q)$ for its Poincaré series. Lusztig has shown that the quotient $P_W(q^2)/P_W(q)$ is equal to a certain power series $L_{W}(q)$, defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in $W$. The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization $L^J_W(s,q) \in \mathbb{Z}[[s,q]]$ depending on a subset $J\subset S$. This new power series specializes to $L_W(q)$ when $s=-1$ and is given explicitly by a sum of r…
Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon
本文通过多面体的$h^*$-多项式给出了带符号多重集排列上主指标与下降统计量联合分布的生成多项式公式，并建立了A、B型广义欧拉数与多面体之间的联系，进而将结果推广到着色排列。
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.
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ${}_2ϕ_1$. This identity is then applied to obtain a generalization of the generating function version of Andrews' f…
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite…
In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such as piecewise polynomaility, while the quantitative properties of these two numbers are quite different. We consider real analogues of monotone double Hurwitz numbers and study the asymptotics for these real analogues. The key ingredient is an interpretation of real tropical covers with arbitrary splittings as factorizations in the symmetric group which genera…
On a double series
针对给定的含参数x、y、w的二重级数展开研究，得到该级数的一项特定函数性质，属于组合数学领域的级数相关研究。
We obtain a new $q$-analogue of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=π/4$, namely \begin{equation*} \sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}}, \end{equation*} where $q$ is a complex number with $|q|<1$. We also show that the Zeilberger-type series $\sum_{k=1}^\infty(3k-1)16^k/(k\binom{2k}k)^3=π^2/2$ has two $q$-analogues with $|q|<1$, one of which is $$\sum_{n=0}^\infty q^{n(n+1)/2} \frac {1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \frac{(q^2;q^2…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfl…
Recently, the concept of a D-analogue was introduced by the author. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series. we consider the D-analogues of the q-binomial coefficients, and a D-analogue of the q-Kummer (Bailey-Daum) sum.
Recently, R. Tauraso established finite $p$-analogues of famous Apéry series for $ζ(2)$ and $ζ(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Apéry-type series for $ζ(4)$ and $ζ(5).$ We also prove a $p$-analogue of Zeilberger's series for $ζ(2)$ confirming a conjecture of Z. W. Sun.
We prove some supercongruences for the truncated hypergeometric series.
A Weighted Sum Formula for Double Eisenstein Series
研究双艾森斯坦级数的加权和公式，通过作者与田坂证明的受限双洗牌关系得出双艾森斯坦级数恒等式，用组合方法及生成级数证明除数和恒等式，实现对相关恒等式的证明。
Positive Weighted Partitions Generated by Double Series
本文研究生成函数为二重级数的加权整数分拆，通过建立 $q$-二重级数的闭式公式证明其系数非负，进而导出整数分拆间的不等式。
We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.
In this short note, we provide an explicit formula to compute every colored double Tornheim's series by using double polylogarithm values at roots of unity. When the colors are given by $\pm 1$ our formula is different from that of Tsumura [On alternating analogues of Tornheim's double series II, Ramanujan J. 18 (2009), 81-90] even though numerical data confirm both are correct in almost all the cases. This agreement can also be checked rigorously by using regularized double shuffle relations of the alternating double zeta values in weights less than eight.
During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.
In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection …
We establish a $q$-analogue of the Bailey-Borwein-Bradley identity generating accelerated series for even zeta values and prove $q$-analogues of Markov's and Amdeberhan's series for $ζ(3)$ using the $q$-Markov-WZ method.
Generalized $n$-series and de Rham complexes
本文研究了交换环上广义$n$-级数的代数与组合性质，证明了经典组合恒等式的$s$-类似物，并定义了导数的$s$-类似物以证明仿射直线上的Poincaré引理与Cartier同构。
A note on a generalized double series
本文利用围道积分推导出涉及 Hurwitz-Lerch zeta 函数的广义双重有限级数的闭式公式，并求出其用特殊函数表示的特例。
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and the other is regarded as $p$-adic analogue of an identity due to Ramanujan.
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We use Young's raising operators to introduce and study double eta polynomials, which are an even orthogonal analogue of Wilson's double theta polynomials. Our double eta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of even orthogonal Grassmannians, and specialize to the single eta polynomials of Buch, Kresch, and the author.
$q$-analogues of $π$-formulas due to Ramanujan and Guillera
本文提出EKHAD-规范化方法，基于$q$-WZ对证明拉马努金$\frac{1}{\pi}$公式和Guillera $\frac{1}{\pi^2}$级数的多个新$q$-模拟，并扩展超几何展开。
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ism…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。