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Abstract In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions ν s , q ( z ) = ( z − 1 2 [ s ] q ) ϱ $\nu _{s,q}(z)=\Bigl(z- \frac{1}{2[s]_{q}}\Bigr)_{\varrho}$ on [ 0 , ∞ ) $[0, \infty )$ , we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.

In this article, we define three-dimensional q-Riordan arrays and q-Riordan representations for these arrays. Also, we give four cases of infinite multiplication three-dimensional matrices of these arrays. As applications, we obtain three-dimensional qq-Pascal-like matrix and its inverse matrix by Heine’s binomial formula, using combinatorial identities. Finally, we consider the generalization of three-dimensional qq-Pascal-like matrix and give some identities involving qq-binomial coefficients.

This article studies the bidimensional versions of four number sequences: balancing, Lucas-balancing, Lucas-cobalancing and cobalancing. Some of the identities and generating functions were presented.

In this paper, the distribution function \[ \varphi (u)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }{\sum \limits_{i=1}^{n}}({X_{i}}-\kappa )\lt u\Bigg),\] and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

We consider two specific families of binomial trees and forests: simply generated binomial d-ary trees and forests versus their increasing phylogenetic version, with tree nodes in increasing order from the root to any of its leaves. The analysis (both pre-asymptotic and asymptotic) consists of some of the main statistical features of their total progenies. We take advantage of the fact that the random distribution of those trees are obtained while weighting the counts of the underlying combinatorial trees. We finally briefly stress a rich alternative randomization of combinatorial trees and forests, based on the ratio of favorable count outcomes to the total number of possible ones.

In this paper, we define (k, l)-Gadovan numbers. We give the Binet-like formula, the generating functions, the exponential generating function of the (k, l)-Gadovan numbers. Also, we derive Cassinilike identity, Catalan-like identity, Vajda-like identity, Honsberger-like identity and D'ocagne-like identity for the (k, l)-Gadovan numbers.

Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author’s background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the method of undetermined coefficients and several others. Among these, the generating function approach is particularly prevalent in enumerative combinatorics due to its versatility and widespread use. The primary objective of this work is to introduce an alternative representation grounded in the theory of Riordan arrays. This representation provides a general formula expressed in terms of the vectors of constants and initial conditions associated with any recurrence relation of a given order, offering a new perspective on the structure of such sequences.

In this paper, using the symmetrizing operator δe1e22−l, we derive new generating functions of the products of p,q-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials.

In this paper, we consider the neutrosophic generalized Rayleigh distribution (NGRD). Various neutrosophic properties of NGRD are developed and discussed. The developed distribution is specifically more useful to model indeterminate data which are skewed lifetime data. Also, the neutrosophic parameters are estimated using the well-known method of maximum likelihood (ML) estimation based on a neutrosophic environment. A simulation study is carried out to establish the achievement of the estimated neutrosophic parameters. As a final point, the proposed NGRD applications in the real world have been discussed with the help of real data. A comparative studies also carried out with some recent proposed neutrosophic distributions.

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials.

This paper aims to give generating functions for the new family of polynomials, which are called parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials by using Golden calculus. Numerous properties of these polynomials with their generating functions are investigated. These generating functions give us a generalization of some well-known generating functions for special polynomials such as Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials. Using the Golden differential operator technique, the functional equation method for generating function, we present some properties of these newly established polynomials.

Background The creation of developing new generalized classes of distributions has attracted applied and theoretical statisticians owing to their properties of flexibility. The development of generalized distribution aims to find distribution flexibility and suitability for available data. In this decade, most authors have developed classes of distributions that are new, to become valuable for applied researchers. Methods This study aims to develop the odd log-logistic generalized exponential distribution (OLLGED), one of the lifetime newly generated distributions in the field of statistics. The advantage of the newly generated distribution is the heavily tailed distributed lifetime data set. Most of the probabilistic properties are derived including generating functions, moments, and quantile and order statistics. Results Estimation of the model parameter is done by the maximum likelihood method. The performance of parametric estimation is studied through simulation. Application of OLLGED and its flexibilities is done using two data sets and while its performance is done on the randomly simulated data set. Conclusions The application and flexibility of the OLLGED are ensured through empirical observation using two sets of lifetime data, establishing that the proposed OLLGED can provide a better fit in comparison to existing rival models, such as odd generalized log-logistic, type-II generalized log-logistic, exponential distributions, odd exponential log-logistic, generalized exponential, and log-logistic.

We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program.

Analyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier-series expansion of the function (1−2αcosθ+α2)−s, we obtain a family of new expansions and generating functions for the Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials Tn and the Gegenbauer polynomials Cn(2) is given.

A variation of ordered trees, where each rightmost edge might be marked or not, if it does not lead to an endnode, is investigated. These marked ordered trees were introduced by E. Deutsch et al. to model skew Dyck paths. We study the number of deepest nodes in such trees. Explicit generating functions are established and the average number of deepest nodes, which approaches 53{5 \over 3} when the number of nodes gets large. This is to be compared to standard ordered trees where the average number of deepest nodes approaches 2.

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0).

The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions.

In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in $ L_{1} $ and $ L_{\infty} $ spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete $ \tau $-Gauss, confluent and Appell's hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete $ \tau $-hypergeometric functions have been derived.

Assemble-to-order (ATO) systems, which keep inventory only for components and assemble the end products after demand is realized, have become prevalent in many industries. We study multi-products and multi-components ATO systems with stochastic lead times. Independent Base Stock (IBS) policies and First-Come-First-Serve (FCFS) allocation rules are used to control component inventories. To find the optimal inventory control parameters for such systems, it is necessary to first characterize the relationship between the system performance measures such as order-based backorders and order fulfillment rate and the inventory control parameters such as reorder points. Most previous studies have focused on ATO systems where the demands are homogeneous Poisson processes. In this paper, closed-form expressions are derived for the relationship between the above performance measures and inventory control parameters where the demands are non-homogeneous. We also examine the impact of common component scarcity and demand variability. Finally, the above conclusions are illustrated by numerical examples.

The purpose of this article is to introduce an extension of Exton's hypergeometric function K16 by using the extended beta function given by Özergin et al. [11]. Some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas are obtained for this extended function. Some special cases of the main results of this paper are also considered.

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

An M/M/1 fluid queue with various vacations is studied in the context of a multi-phase random environment. When the system is in operation (i = 1, 2, …, n), it behaves according to the M/M/1 fluid queue model. However, in any other situation, the system is on vacation, so this leads it to transition into the vacation phase (i = 0). This transition occurs only when there is no data in the system. If the system returns from a vacation and finds it still empty of jobs, it will initiate a new vacation and continue in this pattern until jobs become available in the system, at which point it resumes working. When the vacation phase ends, the probability of the system transitioning to the operational phase is denoted as qi(i = 1, 2, …, n). Subsequently, we derive the stationary probability and analyze the buffer content in relation to the modified Bessel function of the first kind. We utilize the generating function approach and the Laplace–Stieltjes transform to achieve this, enabling us to accomplish our objectives. We provide numerical results to elucidate the overall behavior of the system under consideration.

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

In this paper we revisit our previous study of the local theory of prescribed Jacobian equations associated with generating functions, which are extensions of cost functions in the theory of optimal transportation. In particular, as foreshadowed in the earlier work, we provide details pertaining to the relaxation of a monotonicity condition in the underlying convexity theory and the consequent classical regularity. Taking advantage of recent work of Kitagawa and Guillen, we also extend our classical regularity theory to the weak form A3w of the critical matrix convexity conditions.

The robustness of complex networks was one of the first phenomena studied after the inception of network science. However, many contemporary presentations of this theory do not go beyond the original papers. Here we revisit this topic with the aim of providing a deep but didactic introduction. We pay attention to some complications in the computation of giant component sizes that are commonly ignored. Following an intuitive procedure, we derive simple formulas that capture the effect of common attack scenarios on arbitrary (configuration model) networks. We hope that this easy introduction will help new researchers discover this beautiful area of network science.

In this paper, we give the generating functions of binary product between 2-orthogonal Chebyshev polynomials and kFibonacci, k-Pell, k-Jacobsthal numbers and the other orthogonal Chebyshev polynomials.

In this paper, we define and study the bivariate complex Fibonacci and Lucas polynomials. We introduce a operator in order to derive some new symmetric properties of bivariate complex Fibonacci and bivariate complex Lucas polynomials, and give the generating functions of the products of bivariate complex Fibonacci polynomials with Gaussian Fibonacci, Gaussian Lucas and Gaussian Jacobsthal numbers, Gaussian Pell numbers, Gaussian Pell Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions of the products of bivariate complex Fibonacci polynomials with Gaussian Jacobsthal, Gaussian Jacobsthal Lucas polynomials and Gaussian Pell polynomials.

Abstract In this paper, we deduce the generalized q-difference equations for general Al-Salam–Carlitz polynomials and generalize Arjika’s recent results (Arjika in J. Differ. Equ. Appl. 26:987–999, 2020). In addition, we obtain transformational identities by the method of q-difference equation. Moreover, we deduce U ( n + 1 ) $U(n+1)$ type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials by q-difference equation.

В настоящей работе для вероятности продолжения критических ветвящихся случайных процессов получено асимптотическое разложение в предположении о существовании факториальных моментов λk при k = 5, 6, . . . , m, m < ∞.

Abstract In the present paper, we aim to extend the Hurwitz–Lerch zeta function Φ δ , ς ; γ ( ξ , s , υ ; p ) $\varPhi _{\delta ,\varsigma ;\gamma }(\xi ,s,\upsilon ;p)$ involving the extension of the beta function (Choi et al. in Honam Math. J. 36(2):357–385, 2014). We also study the basic properties of this extended Hurwitz–Lerch zeta function which comprises various integral formulas, a derivative formula, the Mellin transform, and the generating relation. The fractional kinetic equation for an extended Hurwitz–Lerch zeta function is also obtained from an application point of view. Furthermore, we obtain certain interesting relations in the form of particular cases.

This article presents methods and algorithms for the computation of isogenies of degree ℓn. Some of these methods are obtained using recurrence equations and generating functions. A standard multiplication based algorithm for computation of isogeny of degree ℓn has time complexity equal to O(n2 M (n log n)), where M(N) denotes the cost of integers of size N multiplication. The memory complexity of this algorithm is equal to O (n log (n log (n))). In this article are presented algorithms for: determination of optimal strategy for computation of degree ℓn isogeny, determination of cost of optimal strategy of computation of ℓn isogeny using solutions of recurrence equations, determination of cost of optimal strategy of computation of ℓn isogeny using recurrence equations, where optimality in this context means that, for the given parameters, no other strategy exists that requires fewer operations for computation of isogeny. Also this article presents a method using generating functions for obtaining the solutions of sequences (um) and (cm) where cm denotes the cost of computations of isogeny of degree ℓum for given costs p; q of ℓ-isogeny computation and ℓ-isogeny evaluation. These solutions are also used in the construction of the algorithms presented in this article.

Introduction Tries (from retrieval) are one of the most practical data structures with a tree construction in computer science. Tries store string data in leaves of tree. They are often used to store such data so that future retrieval can be made efficient. For example, tries are widely used in algorithms for automatically correcting words in texts. The number of nodes of the same type, which are at the same distance from the root of a rooted tree, is called profile. The analysis of the profile of a tree is of great importance. Because many of the parameters of a rooted tree can be expressed in terms of its profile. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. A generalized version of trie is called bucket trie where each leaf or bucket has a storage capacity of more than one string. Random trie is obtained by defining a random grow rule for trie. We present a detailed study of the limit behavior of the profiles in a random bucket tries. Material and methods In this paper, the methods we apply to derive recurrences satisfied by the expected profiles and to solve them asymptotically for all possible ranges of the distance from the root, are based on the use of Poissonization, Mellin transform, recurrence equations, generating functions, singularity analysis and saddle-point method. Results and discussion Here, as the number of stored strings in a random bucket trie increases, we investigate the asymptotic expectation, variance and limiting distribution of each of the two internal and bucket profiles (i.e. the number of bucket nodes or leaves, and the number of internal nodes or non-leaves which are at the same distance from the root) in random bucket tries. Both the expectation and variances of the two profiles contain periodic functions, and we show those periodic functions are not zero that this point has not been proven in the paper on the ordinary trie. Also, we examine the amount of asymptotic ratio of the expectations of the bucket and internal profiles. Conclusion In this research, we will generalize the most important part of the results for the ordinary tries and also provide a proof for an unproven point in those results. More precisely, the purpose of this article is to study the internal profile (the number of non-leaf nodes at distance from the root) and bucket profile (the number of buckets at distance from the root) in an important data structure called random bucket trie (random trie based on data and maximum capacity of data per each leaf). By some methods in complex analysis, we have shown that for every and some fixed constants and , if is in the range , then the expectations and variances of both profiles contain non-zero periodic functions (the non-zero property of these periodic functions has not been proved for the ordinary tries up to now). We also provide a graph of those periodic functions for by MAPLE. Then we study the ratio of the two profiles for Finally, by finding the asymptotic expansions of Poisson generating functions for the probability generating functions of profiles and then using the Cauchy integral formula, we obtain the asymptotic expansions for the probability generating functions, which indicate that the limiting distributions of profiles are normal. ./files/site1/files/62/3Abstract.pdf

In previous works, some relations between Pringsheim continued fractions and vertices of the paths of minimal length on the suborbital graphs $\mathrm{\mathbf{F}}_{u,N}$ were investigated. Then, for special vertices, the relations between these vertices and Fibonacci numbers were examined. On the other hand, Koshy studied relation between recurrence relations of Fibonacci numbers, Pell numbers and generating functions. In this work, it is showed that every vertex on the path of minimal length of suborbital graph $\mathrm{\mathbf{F}}_{u,N}$ has a Pringsheim continued fraction. Then, by Koshy's motivation, the generating function of the recurrence relation of these pringsheim continued fractions are examined.

Here, our aim is to demonstrate some formulae of generalization of the extended hypergeometric function by applying fractional derivative operators. Furthermore, by applying certain integral transforms on the resulting formulas and develop a new futher generalized form of the fractional kinetic equation involving the generalized Gauss hypergeometric function. Also, we obtain generating functions for generalization of extended hypergeometric function..

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

In this review paper, our aim is to study the current research progress of q-difference equations for generalized Al-Salam–Carlitz polynomials related to theta functions and to give an extension of q-difference equations for q-exponential operators and q-difference equations for Rogers–Szegö polynomials. Then, we continue to generalize certain generating functions for Al-Salam–Carlitz polynomials via q-difference equations. We provide a proof of Rogers formula for general Al-Salam–Carlitz polynomials and obtain transformational identities using q-difference equations. In addition, we gain U(n+1)-type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials via q-difference equations. Finally, we derive two extensions of the Andrews–Askey integral via q-difference equations.

We introduce an explicit form of the connection coefficients for Appell polynomial sequences via Toeplitz-Hessenberg matrix determinants. Generalizing, we give an explicit form of the connection coefficients for arbitrary polynomial sequences and constitute the combinatorial meaning of both constants in terms of integer composition.

We introduce an explicit form of the connection coefficients for Appell polynomial sequences via Toeplitz-Hessenberg matrix determinants. Generalizing, we give an explicit form of the connection coefficients for arbitrary polynomial sequences and constitute the combinatorial meaning of both constants in terms of integer composition.