Krishnaswami Alladi, Alexander Berkovich
In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.
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Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli
本文构造含参数 $L,M$ 的三变量多项式恒等式，统一推广并在极限下导出 Schur 与 Lebesgue 关键恒等式，还给出 Alladi–Schur 精细化及连通 Euler、Lebesgue、Capparelli 定理的新无限层级。
We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…
Affine Jacobi-Trudi Identities and $q,t$-Rogers-Ramanujan Identities
猜想正交和辛Schur函数的对偶Jacobi-Trudi恒等式的仿射类比，用于推导Rogers-Ramanujan恒等式的t-模拟，包括经典、Andrews-Gordon及GOW恒等式，并证明任意高度矩形分区Schur函数的仿射对偶Jacobi-Trudi恒等式。
We consider the family of nearest neighbour interacting particle systems on $\mathbb{Z}$ allowing $0$, $1$ or $2$ particles at a site. We parametrize a wide subfamily of processes exhibiting product blocking measure and show how this family can be "stood up" in the sense of Balázs and Bowen (2018). By comparing measures we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a $2$-repetition condition. By specialising to specific processes we produce two variable identities that are shown to relate to Jacobi triple product and vari…
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.
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…
Character expansions and affine Jacobi-Trudi identities
本文通过给出B_n、C_n和BC_n型多参数Hall--Littlewood多项式在矩形形状下的特征展开，证明了Ole Warnaar提出的六个仿射Jacobi-Trudi型恒等式。
Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$
本文通过BGG范畴O中的张量积重数重新证明斜Schur多项式的Jacobi-Trudi恒等式，并确立截断展开的Schur正性，同时解释系数为张量积重数。
Truncated Theta Series Related to the Jacobi Triple Product Identity
本文证明了Merca关于雅可比三重积截断恒等式猜想的前三种情况及相关恒等式，提出新组合解释，并用组合方法证明了可简化为已知猜想的恒等式。
In this paper we generalize the famous Jacobi's triple product identity, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the results and methods developed in our previous paper math.AG/0310085 several generalizations to Siegel modular forms are obtained. These generalizations are identities satisfied by theta functions with characteristics and their derivatives at zero. Equating the coefficients of the Fourier expansion of these relations to zero yields non-trivial combinatorial identities.
Orbits of Jacobi forms and Theta relations
本文提出Jacobi形式轨道的新方法，用于生成Jacobi theta函数恒等式，简化已知theta关系证明并推广Riemann theta关系。
By considering a limiting form of the q-Dixon_4ϕ_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 Goellnitz's (Big) theorem due to Alladi, and this leads to a two parameter extension of Jacobi's triple product identity for theta functions. Finally, refinements of certain modular identities of Alladi connected to the Goellnitz-Gordon series are shown to follow from a limiting form of the q-Dixon_4ϕ_3 summation.
In the present paper, we introduce the concepts of Jacobi polynomials and intersection enumerators of codes over $\mathbb{F}_q$ and $\mathbb{Z}_{k}$ for arbitrary genus $g$. We also discuss the interrelation among them. Finally, we give the MacWilliams type identities for Jacobi polynomials.
In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the …
Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose 2m}-\frac{2mn}{m+n}{m+n\choose m}^2. $$ We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials.
The Jacobi-Trudi formula implies some interesting quadratic identities for characters of representations of $gl_n$. Earlier work of Kirillov and Reshetikhin proposed a generalization of these identities to the other classical Lie algebras, and conjectured that the characters of certain finite-dimensional representations of $U_q(g-affine)$ satisfy it. Here we use a positivity argument to show that the generalized identities have only one solution.
Formalized $q$-series: The Rogers-Ramanujan Identities and Beyond
本文在 Lean 证明助手中形式化 $q$-级数理论，通过构造 $q$-Pochhammer 符号等基本构件，验证了 Jacobi 三重积公式和 Rogers-Ramanujan 恒等式，为模形式等领域的未来形式化奠定基础。
Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality relation for the big $q$-Jacobi polynomials is also given. A simple evaluation of the Nassrallah-Rahman integral is derived by using this summation formula. A new $q$-beta integral formula is established, which includes the Nassrallah-Rahman integral as a special case. The $q$-summation formula also allows us to recover several strange $q$-series identities.
We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grot…
Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1 summation from the q-Pfaff-Saalschuetz summation. Further, we apply the same method to our previous q-Abel-Rothe summation to obtain, for the first time, Abel-Rothe type generalizations of Jacobi's triple product identity. We also give some results for multiple series.
The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi type continued fractions (J-fractions) that formally enumerate the divergent ordinary generating functions of these sequences. The more general definitions of these J-fractions extend the known expansions of the continued fractions originally proved by Flajolet that generate the rising factorial function, or Pochhammer symbol, $(x)_n$, at any fixed non-zero indeterminate $x \in \mathbb{C}$. The rational convergents of these generalized J-fractions provide formal power series approximati…
In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We introduce the notion of Chebyshev derivations of the first and second kinds, which is based on the polynomial algebra, and corresponding specific differential operators. We derive the elements of their kernels and prove that any element of the kernel of the derivations defines a polynomial identity satisfied by the Chebyshev polynomials of the first and second kin…
Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions
本研究将双变量乘积恒等式的级数展开视为拟周期亚纯函数向量空间的典范基表示，发现了多组新的乘积与Rogers-Ramanujan型和恒等式，揭示了乘积恒等式与Rogers-Ramanujan恒等式间的对应关系并给出应用。
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