A motivated proof of the Göllnitz–Gordon–Andrews identities

We present what we call a “motivated proof” of the Göllnitz–Gordon–Andrews identities. A similar motivated proof of the Rogers–Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized to Gordon’s identities by J. Lepowsky and M. Zhu. We anticipate that the present proof of the Göllnitz–Gordon–Andrews identities will illuminate certain twisted vertex-algebraic constructions.     

The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion. 2000 Mathematics Subject Classification Primary—17B69, 39A13 S. Capparelli gratefully acknowledges partial support from MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca). J. Lepowsky and A. Milas gratefully acknowledge partial support from NSF grant DMS-0070800.     

Harmonic number identities via the Newton–Andrews method

By means of the Bell polynomials, we establish explicit expressions of the higher-order derivatives of the binomial coefficient \(\binom{x+n}{m}\) and its reciprocal \(\binom{x+n}{m}^{-1}\) , and extend the application field of the Newton–Andrews method. As examples, we apply the results to the Chu–Vandermonde–Gauss formula and the Dougall–Dixon theorem and obtain a series of harmonic number identities. This paper generalizes some works presented before and provides a way to establish infinite harmonic number identities.     

A motivated proof of Gordon’s identities

We generalize the ??motivated proof?? of the Rogers?CRamanujan identities given by Andrews and Baxter to provide an analogous ??motivated proof?? of Gordon??s generalization of the Rogers?CRamanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

A generalized Neyman–Pearson lemma for g-probabilities

This paper is concerned with hypothesis tests for g-probabilities, a class of nonlinear probability measures. The problem is shown to be a special case of a general stochastic optimization problem where the objective is to choose the terminal state of certain backward stochastic differential equations so as to minimize a g-expectation. The latter is solved with a stochastic maximum principle approach. Neyman–Pearson type results are thereby derived for the original problem with both simple and randomized tests. It turns out that the likelihood ratio in the optimal tests is nothing else than the ratio of the adjoint processes associated with the maximum principle. Concrete examples, ranging from the classical simple tests, financial market modelling with ambiguity, to super- and sub-pricing of contingent claims and to risk measures, are presented to illustrate the applications of the results obtained.     

Arc spaces and the Rogers–Ramanujan identities

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations, we obtain a new approach to the classical Rogers–Ramanujan Identities. The linking object is the Hilbert–Poincaré series of the arc space over a point of the base variety. In the case of the double point, this is precisely the generating series for the integer partitions without equal or consecutive parts.     

Dougall–Dixon formula and harmonic number identities

The Dougall–Dixon summation formula is reformulated in terms of binomial sums. By computing their second derivatives, we establish several harmonic number identities.      

Rank–Crank-type PDEs and generalized Lambert series identities

We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.     

The curve selection lemma and the Morse–Sard theorem

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C ^r function , we have

A nonlinear Klein–Gordon equation on Kerr metrics

We consider the nonlinear Klein–Gordon equation □u+m²u+λ|u|²u=0, with λ⩾0, outside a Kerr black hole. We solve the global Cauchy problem for large data with minimum regularity. Then, using a Penrose compactification, we prove, in the massless case, the existence of smooth asymptotic profiles and Sommerfeld radiation conditions, at the horizon and at null infinity, for smooth solutions.     

A scale-invariant Klein–Gordon model with time-dependent potential

The goal of the present paper is to derive statements about energy estimates as well as L ^p ?L ^q decay estimates for a Klein?CGordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein?CGordon models with time-dependent mass.     

Dedekind–Mertens lemma and content formulas in power series rings

Let $R?X?$ be the power series ring over a commutative ring R with identity. For $f\in R?X?$, let $A_f$ denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if $g\in R?X?$ such that $A_g$ is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely $A_f^2{A}_g=A_f{A}_{fg}$ for all $f\in R?X?$. More generally for a Prüfer domain R, we prove the content formula ${(A_f{A}_g)}^2=(A_f{A}_g)A_{fg}$ for all $f,g\in R?X?$. As a consequence it is shown that an integral domain R is completely integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R?X?$, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R[X]$, where $R[X]$ is the polynomial ring over R.For a ring R and $g\in R?X?$, if $A_g$ is not locally finitely generated, then there may be no positive integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. Assuming that the locally minimal number of generators of $A_g$ is $k+1$, Epstein and Shapiro posed a question about the validation of the formula $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of $A_g$ is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of $A_g$ is $k+1$, then $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. As a consequence we show that if $A_g$ is finitely generated (in particular if $g\in R[X]$), then there exists a nonnegative integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$.     

Combinatorial proofs of the Newton–Girard and Chapman–Costas-Santos identities

In this paper we give combinatorial proofs of some well known identities and obtain some generalizations. We give a visual proof of a result of Chapman and Costas-Santos regarding the determinant of sum of matrices. Also we find a new identity expressing the permanent of sum of matrices. Besides, we give a graph theoretic proof of the Newton–Girard identity in a generalized form.