Piotr Sniady
We prove a family of new identities fulfilled by multinomial coefficients, which were conjectured by Dykema and Haagerup. Our method bases on a study of the, so-called, triangular operator T by the means of the free probability theory.
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Identities that represent powers of positive integers using multinomial coefficients
本文利用多项式系数表示正整数幂的恒等式，不依赖于多项式定理和多项式范德蒙德卷积。
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.
A few years ago Foda, Quano, Kirillov and Warnaar proposed and proved various finite analogs of the celebrated Andrews-Gordon identities. In this paper we use these polynomial identities along with the combinatorial techniques introduced in our recent paper to derive Garrett, Ismail, Stanton type formulas for two variants of the Andrews-Gordon identities.
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with compositions of Bell polynomials.
We obtain a family of new combinatorial identities for symmetric formal power series.
Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.
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.
A probabilistic approach is provided to establish new hypergeometric identities. It is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime.
We define two classes of representations of quivers over arbitrary fields, called monomorphic representations and epimorphic representations. We show that every representation has a unique maximal nilpotent subrepresentation and the associated quotient is always monomorphic, and every representation has a unique maximal epimorphic subrepresentation and the associated quotient is always nilpotent. The uniquenesses of such subrepresenations imply two identities in the Ringel-Hall algebra. By applying Reineke's integration map, we obtain two identities in the corresponding quantum torus.
Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers
本文引入了随机递归树、中国餐馆过程等组合概率经典对象的多项式与 $r$-变体，利用 Stirling、Eulerian 和 Lah 数的 $r$-版本给出精确分布公式，提供概率解释，证明分布极限定理，并定义分析了 $(r,s)$-Lah 分布。
There is an asymptotic relationship between the multiplicative relations among multinomial coefficients and the (additive) recurrence property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a $1$-cocycle condition in \emph{information cohomology}, an algebraic invariant of phesheaves of modules on \emph{information structures} (categories of observables). Baudot and Bennequin introduced this cohomology and proved that Shannon entropy represents the only nontrivial cohomology class in degree $1$ when the c…
A Lie theoretic interpretation is given for some formulas of Schur functions and Schur $Q$-functions. Two realizations of the basic representation of the Lie algebra $A^{(2)}_2$ are considered; one is on the fermionic Fock space and the other is on the bosonic polynomial space. Via the boson-fermion correspondence, simple relations of the vacuum expectation values of fermions turn out to be algebraic relations of Schur functions.
Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities
本文研究了由求和-乘积恒等式产生的两个划分类中t-hooks数量的不等式，通过生成函数和渐进行式证明了t=1,2时的不等式。
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers-Ramanujan type, involving generating functions, for a cla…
Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_ν(a,b;q)=\sum_{\substack{λ\\[1pt] λ_1\leq a}} q^{c|λ|} P_{2λ}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congru…
Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers
本文通过推广 Roehrig 和 Zwegers 的不定 theta 函数理论，证明若干幂级数为模形式，并由此导出三角数生成函数幂的恒等式，进而揭示其与仿射李超代数分母恒等式的联系。
We give an interpretation of the boson-fermion correspondence as a direct consequence of Jacobi-Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi-Trudi identity the action of Clifford algebra on polynomial algebras that arrives as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions asso…
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{Φ_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{Φ_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $Φ_…
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 Kalita.
We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Le and Zagier.
On Partition Classes Arising from Parity, Differences, and Repeated Smallest Parts
本文研究了与奇偶性、差值和重复最小部分相关的划分类，推导了相关身份并扩展了欧拉划分定理，同时给出了组合和q-级数的证明。
In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property of the recursive sequences. In particular cases we also establish a direct bijection. Besides, we prove further equalities that provide a representation of the sum of $(r+1)$-st and $(r-1)$-st Fibonacci number as the sum of powers of the golden ratio. Similarly, we show a class of natural numbers represented as the sum of powers of the silver ratio.
We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\fracπ{2})$, the complete elliptical integral of the first kind, and its derivatives evaluated at $k=\frac{1}{2}$. These $q$-analogues interpolate those $n$th derivative evaluations by extending $n$ to a complex variable $s$, and we prove that they can be expressed as an infinite product.
The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341.
Explicit Identities and new results for Infinite Series associated with the Ratio of Central Binomial Coefficients
研究中心二项式系数比值相关的无穷级数，推导新的显式恒等式和闭式求值，改进先前结果。
Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov-Witten Theory
本研究证明了Gromov-Witten理论中两个与特定对数Calabi-Yau曲面不变量生成级数相关的多元$q$-二项式猜想恒等式，核心工具是基本超几何级数中的Jackson的Pfaff-Saalschütz求和公式的$q$-模拟。
On the growth of friezes via theta functions
研究证明，给定无环有理型簇代数的管中无限frieze的增长系数相同，利用theta函数的恒等式，扩展了之前在ADE型的结果。
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