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We construct a canonical basis for the space of weakly holomorphic Drinfeld modular forms. And we find that the basis elements satisfy a generating function and the duality among coefficients of the basis elements. Moreover we obtain the congruence properties of t-expansion coefficients of these functions under some conditions.

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one. The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function., 32 pages, 2 tables

In the paper, we first define a degenerate Pochhammer symbol by using the degenerate gamma function and investigate its properties. By using the degenerate Pochhammer symbol, we introduce and investigate a degenerate hypergeometric function. We also define a degenerate Sumudu transform and investigate its properties by using degenerate exponential function. Finally, we give certain the integral representations, derivative formulas, integral transforms, factional calculus applications, and generating functions of the degenerate hypergeometric function.

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

Bu makalede Van Der Laan hibrit dizisi tanıtıldı. Bu dizi ile ilişkili Binet benzeri formül, kısmi toplam ve üreteç fonksiyonu elde edildi. Van Der Laan hibrit dizisinin bazı ilginç özellikleri verildi. Son olarak, Van Der Laan hibrit dizisini içeren bir sirkülant matrisin öz değerleri ve determinantı sunuldu.

Determinantal representation for the generating function of plane partitions with fixed volumes of diagonal parts is investigated in limiting cases.

We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j. We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.

We present a novel generating function (GF) method for the self-condensing vinyl polymerization (SCVP) system with any initial distribution of preexisted polymers. Such a method was proven to be especially useful to investigate the semi-batch SCVP system allowing a sequence of feeding operations during the polymerization. Consequently, the number-, weight-, and z-average molecular weights as well as polydispersity index of hyperbranched polymers can be explicitly given, which are determined by predetermined feeding details and conversions in each polymerization step. These analytical results are further confirmed by the corresponding Monte Carlo simulation. Therefore the present GF method has provided a unified treatment on the semi-batch SCVP system. Accordingly, hyperbranched polymers with desired properties can be prepared by designing feeding details and presetting conversions at each step based on the present GF method.

For a graph $\Gamma$, let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

A new class of continuous distributions called the generalized Burr X-G family is introduced. Some special models of the new family are provided. Some of its mathematical properties including explicit expressions for the quantile and generating functions, ordinary and incomplete moments, order statistics and Rényi entropy are derived. The maximum likelihood is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of two applications to real data sets.

We present a generating-function representation of a vector defined in either Euclidean or Hilbert space with arbitrary dimensions. The generating function is constructed as a power series in a complex variable whose coefficients are the components of a vector. As an application, we employ the generating-function formalism to solve the eigenproblem of a vibrating string loaded with identical beads. The corresponding generating function is an entire function. The requirement of the analyticity of the generating function determines the eigenspectrum all at once. Every component of the eigenvector of the normal mode can be easily extracted from the generating function by making use of the Schlafli integral. This is a unique pedagogical example with which students can have a practical contact with the generating function, contour integration, and normal modes of classical mechanics at the same time. Our formalism can be applied to a physical system involving any eigenvalue problem, especially one having many components, including infinite-dimensional eigenstates.

We discuss the optimum communication capacity of bosonic information carriers propagating through a tunnel junction. Based on the multi-contour Keldysh Green function, we evaluate the information generating function, or the Renyi entanglement entropy, of the reduced density matrix subjected to the constraint of local heat quantity in the steady state. The Renyi entanglement entropy of order zero is the partition function, which exponentially depends on the optimum capacity. For the perfect transmission, the self-information and the local heat quantity, i.e., the energy of signals, are perfectly linearly correlated. The water-filling theorem is recovered in the wave-like regime.

We consider the bargraph representation of geometrically distributed words, which we use to define the water capacity of such words. We first find a bivariate capacity generating function for all geometrically distributed words, from which we compute the generating function for the mean capacity. Thereafter, by making extensive use of Rice’s method (Rice’s integrals) we derive an asymptotic formula for the average capacity of random words of length n as n tends to infinity.

Many LED lighting applications involve the design of multiple optical surfaces. A prime example is a single lens with two refractive surfaces. In this paper, we consider an LED light source approximated as a point and a far-field target intensity. Using Hamilton’s characteristic functions, the design problem is converted into two generalized Monge–Ampère equations by deriving a generating function for each optical surface. The generating function is a generalization of the cost function in optimal transport theory. The generalized Monge–Ampère equations are solved using an iterative least-squares algorithm. To compute the first optical surface, we choose an intermediate far-field target intensity. By choosing different intermediate target intensities based on the source and target intensity, we develop a “knob” to distribute the refractive power over both surfaces of the lens. We apply the algorithm on two example problems and show it is capable of producing complicated target distributions.

In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

Exclusive roles have been played by special functions in applied mathematics. It is not astonishing when new classes of special functions are established as the issues associated with special functions are too immense. In this article, the generalized form of hybrid Bessel functions is introduced by using suitable integral transforms and properties of exponential operators. Certain properties including generating function, series definition, operational rule and integral representation of the generalized form of hybrid Bessel functions are derived.

We study a multiplicative controlled branching process which is subjected to an independent immigration at each generation. We discuss probabilistic properties such as generating function, mean and...

In this work, we consider the information generating measure and relative information generating measure and develop some associated results concerning mixed systems with independent and identicall...

In this work we present an algebraic approach to the dynamics and perturbation theory at tree-level for gauge theories coupled to matter. The field theories we will consider are: Chern-Simons-Matter, Quantum Chromodynamics, and scalar Quantum Chromodynamics. Starting with the construction of the master action in the classical Batalin-Vilkovisky formalism, we will extract the L∞-algebra that allow us to recursively calculate the perturbiner expansion from its minimal model. The Maurer-Cartan action obtained in this procedure will then motivate a generating function for all the tree-level scattering amplitudes. There are two interesting outcomes of this construction: a generator for fully-flavoured amplitudes via a localisation on Dyck words; and closed expressions for fermion and scalar lines attached to n-gluons with arbitrary polarisations.

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole composition. We achieve this by conceptualizing a generating function that tracks compositions as well as the number of impulses used. This conceptualization is repeated for words (over a finite alphabet) represented by bargraphs.

In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

In this paper, we first define the bi-periodic balancing numbers and quaternions. We give the generating function and Binet formula for this quaternion. Then, we obtain some identities and properties including this quaternion.

Our purpose is to give a generalization of geometric polynomials by applying an appropriate linear transformation on the generalized factorial function. Some identities are investigated including explicit formula, generating function and recurrence relations. Furthermore, some relations with other polynomials are given.

In this paper we introduce a new one parameter generalization of the Pell quaternions – P (r, n)-Pell quaternions. We give some of their properties, among others the Binet formula, convolution identity and the generating function.

We consider a single-line queuing system with an infinite buffer that receives two Poisson flows of customers with different intensities. Customers of the first type have preemptive priority over customers of the second type. In addition, at the time of the end of servicing, a high-priority customer with some probability can drop all low-priority customers in the queue. Serving both types of customers has an exponential distribution with different parameters. We show expressions for calculating stationary probabilities in this system, the probability of servicing a low-priority customer in terms of the generating function, and a formula for the average number of customers of the second type.

In this paper, we introduce bihyperbolic balancing and Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities and the generating function.

In 2013, Baruah and Sarmah, and Xia and Yao independently obtained generating function for the sequences p−8(2n + 1) and p−8(4n + 3), where p−8(n) counts the number of partitions of n in eight colours. In this article, we generalize the identities and as a consequence, establish several Ramanujan type congruences modulo higher powers of 2.

This article proposes a new class of models called the transmuted alpha power-G (TAPO-G) family of distribution for modeling lifetime processes. This class of model extends the well-known existing ...

The Padovan sequence is a sequence similar to the Fibonacci sequence, the former being third order and the latter second. Having several applications in architecture, these numbers are directly related to plastic numbers. In this paper, the Padovan sequence is studied and investigated from the standpoint of linear algebra. With this, we will study the matrix and the generating function of the extensions of this sequence (Tridovan and Tetradovan), thus determining the generalization of this sequence.

In this paper we carry out a reliability study of the systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of structure’s reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, several numerical results are presented and some figures are constructed and appropriately commented.

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation ( ω ) {(\omega)} of the two-dimensional quantum algebra ℰ q ⁢ ( 2 ) {\mathcal{E}_{q}(2)} is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored.

Dan Reznik found, by computer experimentation, a number of conserved quantities associated with periodic billiard trajectories in ellipses. We prove some of his observations using a non-standard generating function for the billiard ball map. In this way, we also obtain some identities valid for all smooth convex billiard tables.

Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials.

We consider a discrete-time queueing system with two queues and one server. The server is allocated in each slot to the first queue with probability $$\alpha $$ and to the second queue with probability $$1-\alpha $$ . The service times are equal to one time slot. The queues have exponentially bounded, but general, arrival distributions. The mathematical description of this system leads to a single functional equation for the joint probability generating function of the stationary system contents. As the joint stochastic process of the system contents is not amenable for exact analysis, we focus on an efficient approximation of the joint probability generating function. In particular, first we prove that the partial probability generating functions, present in the functional equation, have a unique dominant pole. Secondly, we use this information to approximate these partial probability generating functions by truncating an infinite sum. The remaining finite number of unknowns are estimated from a noise perturbed linear system. We illustrate our approach by various numerical examples and verify the accuracy by means of simulation.

Waiting line problems with server vacation have envisaged with increasing complexities and their explicit transient solutions are rigorous in computations, at the same time such solutions are valued for studying the dynamical behaviour of queuing systems over a finite period predominantly utilizes within the state-of-art design process for a real time system. Keeping this fact in mind we adopt continued fractions and generating function to derive explicit expressions for transient state probabilities. In this paper, we consider the waiting line problem with a single server which adopts the multi vacations policy. We analyzed the transient part for a single server multi vacations queue with discouragement . It is also obtained the expected value of the state of the system using stationary queue size distribution, which gives a quick glance of a system performance.

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers–Ramanujan identities, the generating function yields two formulas in Slater’s list. The same formulas were constructed by Hirschhorn. Similar formulas were obtained by Bringmann, Mahlburg and Nataraj. We also use staircases to give alternative triple series for partitions into [Formula: see text]-distinct parts for any [Formula: see text].

В даппой работе получена рекуррентная формула для сечений производящих рядов числа путей па целочисленной решетке и доказано, что сечения производящей функции для решеточных путей являются рациональными функциями (аналог теоремы Муавра).

Motivated by recent investigations, the main object of this paper is to construct the new (p, q)-analogy definitions of the various families of (p, q)-Bessel functions using the generating function...

In this paper we study a geodesic radiating stellar model in which the Einstein field equations contain a cosmological constant and electric charge. Firstly, we study the boundary condition by introducing a generating function that is directly related to the horizon function. This generating function transforms the boundary condition into an algebraic equation which is solvable. New classes of exact solutions can be obtained by specifying a form for the generating function. It is possible to express physical quantities such as the mass and compactness factor in terms of the generating function. We also regain earlier results with only electric charge in this process. Secondly, we transformed the boundary condition into a second order differential equation in terms of the generating function. By specifying certain parameters we solved the boundary condition by direct integration and obtained a new special class of exact solutions in terms of confluent hypergeometric functions.

We study the computation of admissible marking sets in generalized Petri nets. We first show that the admissibility checking in the generalized Petri net is NP-hard. Then, we consider a special subclass of generalized Petri nets called weighted-synchronization-free nets in which each transition has at most one input place. For a net in this subclass, we propose a generating function to compute by dynamic programming the set of admissible markings for a given generalized mutual exclusion constraint .

The present paper deals with the performance evaluation of an M/M/1 retrial queue with collisions, transmission errors and unreliable server. To the best of our knowledge, there are no works that h...

In this paper, we propose a new family of algorithms for underdetermined blind identification based on the second generating function (SGF). For the real case, the proposed algorithm first takes advantage of the partial derivatives of the SGF of the observation signals to construct a tensor and then reconstructs this tensor by grouping the parallel factors. The reconstructed tensor, which expands the length of several dimensions of the original tensor, transforms the underdetermined case into an overdertermined or determined case when $$({M^2} + M)/2 \ge N$$ , where M is the number of sensors and N is the number of sources. Thus, obtaining the estimation of the mixing matrix by decomposing the reconstructed tensor results in the performance improvement. The analysis of the uniqueness of tensor decomposition and the complex analysis show that the proposed algorithm relaxes the limitation of the maximal number of sources and owns lower computational complexity order compared with the conventional ALESCAF algorithm. We also extend the proposed algorithm to the complex case by complex-to-real transformation. Simulation results verify the effectiveness of the proposed algorithms and show that the proposed algorithms are preferred when the signal-to-noise ratio is higher than 5 dB.

Let [Formula: see text] be the generating function for the number [Formula: see text] of spanning trees in the circulant graph Cn(s1, s2, …, sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, …, sk, n) of odd valency. We illustrate the obtained results by a series of examples.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

The paper is concerned with special questions on the behavior of zeros of sequences of Bernstein polynomials. For a piecewise linear generating function, computer mathematics machinery was used to find the rules controlling the limit behavior of zeros as the number of the Bernstein polynomial unboundedly increases. New problems for theoretical investigations are formulated.

This paper proposes Kalman filters to investigate the state estimation of continuous-time linear fractional-order systems disturbed by correlated colored noises. A difference equation is obtained by discretizing the fractional-order differential equation via Tustin generating function. Besides, an augmented vector is determined by the state vector and the colored noise vector to deal with the problems on the colored process noise or the colored measurement noise with fractional-orders. In fact, a more accurate state estimation can be achieved using the discretization method via Tustin generation function for the investigated fractional-order systems, compared with Grünwald-Letnikov difference. Finally, two illustrative examples are provided to validate the effectiveness of the proposed algorithms in this paper.