Narrow Results

In this paper, we introduce and study a spinor algebra of $k$-balancing numbers referred to as the $k$-balancing and $k$-Lucas-balancing spinors. First, we give $k$-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of $k$-balancing numbers by defining a linear and injective correspondence between $k$-balancing and $k$-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations.

Generalized Hecke operators, originating from the replication formula in Monstrous Moonshine, were extended in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J.62(3) (2023) 675–717] to apply to harmonic Maass functions on modular curves of higher genera, building on works in [M. Koike, On replication formula and Hecke operators, Nagoya University, preprint]. Their action was further applied to weakly holomorphic modular functions, deriving numerous arithmetic properties of Fourier coefficients. In this paper, we extend these operators to weakly holomorphic modular forms of arbitrary even non-positive weights. In the process, we show [Proposition 3.1 of L. Beneish and M. H. Mertens, On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras, Math. Z.297(1–2) (2021) 59–80], which is used to obtain dimension formulas for certain vertex operator algebras, can be derived from our results. Additionally, we identify the conditions under which the action of the generalized Hecke operator preserves holomorphicity. Moreover, we show that this action can be expressed as a linear combination of same-level or lower-level forms, refining the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J.62(3) (2023) 675–717]. Finally, we establish more general congruence relations on Fourier coefficients, with the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J.62(3) (2023) 675–717] emerging as a special case.

In this paper we give some undecidability and decidability results about context-free languages. First, we prove that the problem of deciding whether a context-free language which admits a holonomic generating function is Turing equivalent to the finiteness question for r.e. sets. Second, we show that the Equivalence Problem is decidable for a suitable class of languages, called LCL_R.

We consider heap-ordered trees in this paper. The main theorems in this paper show that the average and the variance of the altitude (or level) of the nodes in a random n-node heap-ordered tree are and , respectively where γ=0.5772156649… is the Euler’s constant.

Let S denote a subset of the positive integers, and let p_S(n) be the associated partition function, that is, p_S(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then p_S(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by p_S(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

A point q in a contact manifold is called a translated point for a contactomorphism ϕ with respect to some fixed contact form if ϕ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism ϕ of ℝ²ⁿ⁺¹ or ℝ²ⁿ × S¹ contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if ϕ is positive then there are infinitely many nontrivial geometrically distinct iterated translated points, i.e. translated points of some iteration ϕ^k. This result can be seen as a (partial) contact analog of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of ℝ²ⁿ, and is obtained with generating functions techniques.

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra for the transition matrix A^G of the directed edges of G. The generalized Catalan numbers enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $ℂ{\text{P}}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of $ℂ{\text{P}}^d$ that has at least $d+2$ nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.

We study a M/M/1 queue in a multi-phase random environment, where the system occasionally suffers a disastrous failure, causing all present jobs to be lost. The system then moves to a repair phase. As soon as the system is repaired, it moves to phase i with probability q_i ≥ 0. We use two methods of analysis to study the probabilistic behavior of the system in steady state: (i) via probability generating functions, and (ii) via matrix geometric approach. Due to the special structure of the Markov process describing the disaster model, both methods lead to explicit results, which are related to each other. We derive various performance measures such as mean queue sizes, mean waiting times, and fraction of lost customers. Two special cases are further discussed.

It is known that wormhole geometry could be found solving the Einstein field equations by tolerating the violation of null energy condition (NEC). Violation of NEC is not possible for the physical matter distributions, however, it can be achieved by considering distributions of “exotic matter”. The main purpose of this work is to find generating functions comprising the wormhole-like geometry and discuss the nature of these generating functions. We have used the Herrera et al. [Phys. Rev. D77, 027502 (2008)] approaches of obtaining generating functions in the background of wormhole spacetime. Here we have adopted two approaches for solving the field equations to find wormhole geometry. In the first method, we have assumed the redshift function f(r) and the shape function b(r) and solve for the generating functions. In another attempt, we assume generating functions and redshift functions and then try to find shape functions of the wormholes.

In this paper, we construct the quantum fields of three types of universal characters corresponding to partition shapes $\pi =(3)$, $\pi =(2,1)$ and $\pi =(1^3)$ and derive the commutation relations of quantum fields. By acting the quantum fields on the constant function $1$, the generating functions of three types of universal characters have been investigated. Furthermore, we discuss three integrable systems characterized by $(3)$-type, $(2,1)$-type and $(1^3)$-type universal characters.

The associated Legendre functions for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics .

We define the generalized vertex operators expressed by charged fermions and state vectors corresponding to the n-partition. The action of the generalized vertex operators on state vectors and the generalized universal character (UC) plane partitions have been investigated by the fermion calculus techniques. Furthermore, we construct the infinite scalar product and the generating function associated with the generalized UC plane partitions.

We introduce a family of multiplicative distributions {μ_s|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μ_s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number $b_n$ of minimal varieties of given exponent $n$ is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit $\beta ={lim}_{n\to \infty }\sqrt[n]{b_n}$ exists and can be expressed as the positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank $d$. It follows from classical results on lacunary power series that the generating function of the sequence $b_n$, $n=1,2,\dots$, is transcendental. With the same approach we construct examples of free graded semigroups $〈Y〉$ with the following property. If $d_n$ is the number of elements of degree $n$ of $〈Y〉$, then the limit $\delta ={lim}_{n\to \infty }\sqrt[n]{d_n}$ exists and is transcendental.

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol’d [Lagrange and Legendre cobordisms. I, Funkt. Anal. Prilozhen. 14(3) (1980) 1–13, 96.] — consisting of big Legendrian submanifolds between two smaller ones. We are interested in such cobordisms which fit with generating functions, and wonder which structures and obstructions come with this notion. As a central result, we show that the classes of Legendrian concordances with respect to the generating function equipment can be given a group structure. To this construction, we add one of a homotopy with respect to generating functions.

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking into account genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots with unknotting number one.

Let $N⊴G$ be a pair of finite subgroups of ${\mathrm{\text{SL}}}_2(ℂ)$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant’s generating functions for the decomposition of the ${\mathrm{\text{SL}}}_2(ℂ)$-module $S^k(V)$ restricted to $N◃G$ in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.

In a manner analogous to the definition of Stirling numbers of the first and second kind we define a set of numbers for restricted random walks of finite memory. These two sequences of counting numbers should be useful in the exact enumeration of properties of simple and restricted random walks.