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It is shown how to approximate the solution of functional differential equations in terms of Bell's polynomials. Some numerical checks are shown, by using the computer algebra system Mathematica © . [ABSTRACT FROM AUTHOR]

This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function t m e λ x (t) e λ x (t) - ∑ l = 0 m - 1 (1) l , λ t l l ! = ∑ n = 0 ∞ B n , λ [ m - 1 ] (x) t n n ! , | t | < min { 2 π , 1 | λ | } , λ ∈ ℝ \ { 0 }. We deduce their associated summation formulas and their corresponding determinant form. Also we focus our attention on the zero distribution of such polynomials and perform some numerical illustrative examples, which allow us to compare the behavior of the zeros of degenerate hypergeometric Bernoulli polynomials with the zeros of their hypergeometric counterpart. Finally, using a monomiality principle approach we present a differential equation satisfied by these polynomials. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type polynomials, and explore some of their algebraic properties, including summation formulas and their determinant form. The majority of our results are proven using generating function methods. Additionally, we investigate the monomiality principle related to these polynomials and identify the corresponding derivative and multiplicative operators. [ABSTRACT FROM AUTHOR]

In this note, we prove the last remaining case of the original 15 two-term supercongruence conjectures for sporadic sequences. The proof utilizes a new representation for this sequence (due to Gorodetsky) as the constant term of powers of a Laurent polynomial. [ABSTRACT FROM AUTHOR]

Let f (t) = ∑ n = 0 + ∞ C f , n n ! t n be an analytic function at 0, and let C f , n (x) = ∑ k = 0 n n k C f , k x n - k be the sequence of Appell polynomials, referred to as C-polynomials associated tof, constructed from the sequence of coefficients C f , n . We also define P f , n (x) as the sequence of C-polynomials associated to the function p f (t) = f (t) (e t - 1) / t , called P-polynomials associated tof. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on f, we introduce and study the bivariate complex function P f (s , z) = ∑ k = 0 + ∞ z k P f , k s z - k , which generalizes the s z function and is denoted by s (z , f) . Thirdly, the paper's main contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz's formula, by constructing a novel class of functions defined by L (z , f) = ∑ n = n f + ∞ n (- z , f) , which are intrinsically linked to C-polynomials and referred to as LC-functions associated tof (the constant n f is a positive integer dependent on the choice of f). [ABSTRACT FROM AUTHOR]

A new bidimensional version of cobalancing numbers and Lucas-balancing numbers are introduced. Some properties and identities satisfied by these new bidimensional sequences are studied. [ABSTRACT FROM AUTHOR]

Given d , s ∈ N , a finite set A ⊆ Z and polynomials φ 1 , ⋯ , φ s ∈ Z [ x ] such that 1 ≤ deg φ i ≤ d for every 1 ≤ i ≤ s , we prove that | A (s) | + | φ 1 (A) + ⋯ + φ s (A) | ≫ s , d | A | η s , for some η s ≫ d log s / log log s . Moreover if φ i (0) ≠ 0 for every 1 ≤ i ≤ s , then | A (s) | + | φ 1 (A) ⋯ φ s (A) | ≫ s , d | A | η s. These generalise and strengthen previous results of Bourgain–Chang, Pálvölgyi–Zhelezov and Hanson–Roche-Newton–Zhelezov. We derive these estimates by proving the corresponding low-energy decompositions. The latter furnish further applications to various problems of a sum-product flavour, including questions concerning large additive and multiplicative Sidon sets in arbitrary sets of integers. [ABSTRACT FROM AUTHOR]

Let N be the set of all nonnegative integers. For S ⊆ N and n ∈ N , let the representation function R S (n) denote the number of solutions of the equation n = s + s ′ with s , s ′ ∈ S and s < s ′ . In this paper, we determine the structure of C , D ⊆ N with C ∪ D = [ 0 , m ] , C ∩ D = { r 1 , r 2 } , r 1 < r 2 and 2 ∤ r 1 such that R C (n) = R D (n) for any nonnegative integer n. [ABSTRACT FROM AUTHOR]

A Diophantine m-tuple over a finite field F q is a set { a 1 , ... , a m } of m distinct elements in F q ∗ such that a i a j + 1 is a square in F q whenever i ≠ j . In this paper, we study M(q), the maximum size of a Diophantine tuple over F q , assuming the characteristic of F q is fixed and q → ∞ . By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. [ABSTRACT FROM AUTHOR]

ABSTRACTRecently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let [Formula: see text] be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with [Formula: see text] and the probabilistic degenerate Euler polynomials associated with [Formula: see text]. Also, we intoduce the probabilistic degenerate [Formula: see text]-Stirling numbers of the second associated with [Formula: see text] and the probabilistic degenerate two variable Fubini polynomials associated with [Formula: see text]. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of [Formula: see text], we treat the gamma random variable with parameters [Formula: see text], the Poisson random variable with parameter [Formula: see text], and the Bernoulli random variable with probability of success [Formula: see text].

AbstractThis paper aims to introduce a new class of special polynomials called q-modified Laguerre-Appell polynomials. Some definitions and concepts related to this class of polynomials, including generating function and series definition are explored. Also, certain interesting properties related to this class and some of its members are investigated. Furthermore, the class of 2D q-modified Laguerre-Appell polynomials is introduced. Finally, we present certain graphical representations related to some members of these new classes.

The study of special functions has become an enthralling area in mathematics because of its properties and wide range of applications that are relevant into other fields of knowledge. Developing topics in special functions involves the investigation of Apostol-type polynomials encompassing the combinations, extensions, and generalizations of some classical polynomials such as Bernoulli, Euler, Genocchi, and tangent polynomials. One particular type of these polynomials is the Apostol-Frobenius-Euler polynomials of order aa denoted by Hnα(z;u;λ){H}_{n}^{\alpha }\left(z;\hspace{0.33em}u;\hspace{0.33em}\lambda ). Using the saddle point method, Corcino et al. obtained approximations for the higher-order tangent polynomials. They also established a new method to derive its approximations with enlarged region of validity. In this article, it is found that these methods are applicable to the higher-order Apostol-Frobenius-Euler polynomials. Consequently, approximations of higher-order Apostol-Frobenius-Euler polynomials in terms of the hyperbolic functions are obtained for large values of the parameter nn, and its uniform approximations with enlarged region of validity are also derived. Moreover, approximations of the generalized Apostol-type Frobenius-Euler polynomials of order α\alpha with parameters a,b,a,b, and cc are obtained by applying the same methods. Graphs are provided to show the accuracy of the exact values of these polynomials and their corresponding approximations for some specific values of the parameters.

In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered. [ABSTRACT FROM AUTHOR]

For two real bases q 0 , q 1 > 1 , we consider expansions of real numbers of the form ∑ k = 1 ∞ i k / (q i 1 q i 2 ... q i k) with i k ∈ { 0 , 1 } , which we call (q 0 , q 1) -expansions. A sequence (i k) is called a unique (q 0 , q 1) -expansion if all other sequences have different values as (q 0 , q 1) -expansions, and the set of unique (q 0 , q 1) -expansions is denoted by U q 0 , q 1 . In the special case q 0 = q 1 = q , the set U q , q is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases (q 0 , q 1) with trivial U q 0 , q 1 from those with non-trivial U q 0 , q 1 is the graph of a function G (q 0) that we call generalized golden ratio. Similarly, the curve separating pairs (q 0 , q 1) with countable U q 0 , q 1 from those with uncountable U q 0 , q 1 is the graph of a function K (q 0) that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in q 0 and q 1 , that G and K are continuous, strictly decreasing, hence almost everywhere differentiable on (1 , ∞) , and that the Hausdorff dimension of the set of q 0 satisfying G (q 0) = K (q 0) is zero. We give formulas for G (q 0) and K (q 0) for all q 0 > 1 , using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function x M (x + 1) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

A renewed interest in combinatorial and arithmetic properties as well as applications to differential equations, identities, formulas, and probability theory has been sparked by the study of degenerate versions of several specific numbers and polynomials. The article aims to explore a 3D unified degenerate class of generalized Fubini polynomials by utilizing 2D generalized degenerate polynomials. The potential of applications are provided by deriving certain computational formulas and identities,recurrence relations and derivative expressions for the 3D degenerated Gould–Hopper–Fubini, 3D degenerate Hermite-Fubini and 3D degenerate 2-iterated Fubini polynomials, which are extracted out of the 3D degenerate generalized Fubini polynomials. Finally, the behaviour of zeros of two concrete degenerate polynomials with some specific set of parameters is shown by drawing graphs using Mathematica [ABSTRACT FROM AUTHOR]

The binary signed-digit (BSD) representation of integers is used for efficient integer computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence. In this paper, we show previously unknown connections between BSD integer representations and the Stern polynomial. We then exploit these connections to devise a fast algorithm to count optimal BSD representations on a range of integers and calculate their weights. [ABSTRACT FROM AUTHOR]

We examine the problems associated with the Collatz map T from the point of view of functional analysis. We associate with T a certain linear operator T and show that cycles and (hypothetical) divergent trajectories (generated by T) correspond to certain classes of fixed points of the operator T . We also show the relationship between the dynamic properties of the operator T and the map T. We prove that the absence of non-trivial cycles of T leads to hypercyclicity of the operator T . In the second part, we show that the index of the operator I d - T ∈ L (H 2 (D)) provides an upper estimate for the number of cycles of T. For the proof, we consider the adjoint operator F = T ∗ F : g → g (z 2) + z - 1 3 3 g (z 2 3) + e 2 π i 3 g (z 2 3 e 2 π i 3) + e 4 π i 3 g (z 2 3 e 4 π i 3) , which was first introduced by Berg, Meinardus in [3], and show that it has no non-trivial fixed points in H 2 (D) . Furthermore, we calculate the resolvent of the operator F and derive the equation for the characteristic function of the total stopping time σ ∞ as an application. In addition, we construct an invariant measure for T in a slightly different setup, and investigate how the operator T acts on generalized arithmetic progressions. [ABSTRACT FROM AUTHOR]

In the paper, the authors intrinsically observe that Hoffman’s combinatorial identity discovered in 1992 and Genčev’s combinatorial identities extended in 2024 can be reformulated in terms of complete Bell polynomials. They provide alternative proofs and offer some elementary generalizations of these combinatorial identities. [ABSTRACT FROM AUTHOR]

The probability generating function (pgf) B ⁢ (s) of the limiting conditional law (LCL) of a subcritical Markov branching process (Z t : t ≥ 0) (MBP) has a certain integral representation and it satisfies B ⁢ (0) = 0 and B ′ ⁢ (0) > 0 . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 (Z t = j) have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL. [ABSTRACT FROM AUTHOR]

In the current work, it is constructed the Motzkin matrix obtained by using Motzkin numbers M = (m rs) and is examined the sequence spaces c (M) and c 0 (M) described as the domain of Motzkin matrix M in the spaces c and c0, respectively. It is investigated topological properties, established Schauder basis and stated Köthe duals of the aforementioned spaces. Additionally, it is characterized the matrix classes from c (M) and c 0 (M) to the classical sequence spaces and vice versa. Finally, Motzkin core of any sequence is presented and it is elaborated some inclusion relation of just described new type of core. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish several identities and relations involving q-analogues of the Pochhammer k-symbol. Moreover, the authors generalize several identities and relations for q-analogues of the Catalan numbers and the Catalan–Qi numbers. [ABSTRACT FROM AUTHOR]

In this paper, an expansion of the classical hyperbolic functions is presented and studied. Also, many features of the k - Pell hyperbolic functions are given. Finally, some graph and curved surfaces related to the k − Pell hyperbolic functions are introduced. [ABSTRACT FROM AUTHOR]

A hybrid binary tree is a complete binary tree where each internal node is labeled with 1 or 2, but with no left (1, 1)-edges. In this paper, we consider enumeration of the set of hybrid binary trees according to the number of internal nodes and some other combinatorial parameters. We present enumerative results by giving Riordan arrays, bivariate generating functions, as well as closed formulas. As a consequence, we obtain some new combinatorial matrices, one of which is analogous to the Borel triangle. We also present a bijection between the set of all hybrid binary trees with n internal nodes and the set of generalized Schröder paths from (0, 0) to (2n, 0) which are consist of up steps u = (1 , 1) , horizontal steps h = (2 , 0) , down steps d = (1 , - 1) , and double up steps U = (2 , 2) , and never travel below the x-axis. [ABSTRACT FROM AUTHOR]

In this article, we consider several problems about friable numbers in Piatetski-Shapiro sequences, such as the ternary Goldbach type problem, Diophantine approximation, almost primes, intersections of Piatetski-Shapiro sequences and Beatty sequences, intersections of Piatetski-Shapiro sequences and so on. By using well known properties, we establish two exponential sums involving fractional powers over friable numbers. [ABSTRACT FROM AUTHOR]

Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.

We present a new complex analytic proof of the two classical formulas evaluating the sum of powers of consecutive integers which involve Stirling or Eulerian numbers. Our method generalizes that recently obtained by the second and third author for the sum of squares. [ABSTRACT FROM AUTHOR]

The introduction of two-parameter (p , q) -calculus and Lie algebras in 1991 has spurred a wave of recent research into (p , q) -special polynomials, including (p , q) -Bernoulli, (p , q) -Euler, (p , q) -Genocchi and (p , q) -Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce (p , q) -sine and (p , q) -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as (p , q) -integral representations, (p , q) -partial derivative formulae and (p , q) -addition formulae. It is worth noting that the utilization of the (p , q) -polynomials introduced in this study, along with other (p , q) -polynomials, can lead to the derivation of various identities that differ from the ones presented here. [ABSTRACT FROM AUTHOR]

We obtain explicit upper bounds for the Touchard polynomials T n (x) , for x > 0 . When applied to the Bell numbers B n = T n (1) , such bounds are asymptotically sharp. A simple probabilistic approach based on estimates of moments of nonnegative random variables is used. Applications giving upper bounds for the moments of a certain subset of Jakimovski-Leviatan operators are also provided. [ABSTRACT FROM AUTHOR]

We study polytopes defined by inequalities of the form ∑ i ∈ I z i ≤ 1 for I ⊆ [ d ] and nonnegative z i where the inequalities can be reordered into a matrix inequality involving a column-convex { 0 , 1 } -matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs G with a Hamiltonian path, which we call spinal graphs. We show that the volumes of these flow polytopes are given by the number of upper (or lower) G-cyclic orders defined by the graphs G. As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of k-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's k-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the h ∗ -polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the h ∗ -polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their h ∗ -polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the sequences of the Stirling numbers of the first and second kind with higher level are both Pólya frequency and log-concave. Then, we show that some polynomials related to the above Stirling numbers with higher level are q-log-convex or strongly q-log-convex. Furthermore, we establish that the linear transformation related to the Stirling numbers of the first kind with level 2 preserves the log-convexity. [ABSTRACT FROM AUTHOR]

In this article, we consider compositions of positive integers with 2s and 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2s and 3s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2s and 3s.

It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ\lambda -analogue of the unsigned Stirling numbers of the first kind and that of the rr-Stirling numbers of the first kind. In this article, we introduce a λ\lambda -analogue of the shift algebra (called λ\lambda -shift algebra) and investigate normal ordering in the λ\lambda -shift algebra. From the normal ordering in the λ\lambda -shift algebra, we derive some identities about the λ\lambda -analogue of the unsigned Stirling numbers of the first kind.

In the 1970s, Rota began to build completely rigid foundations for the theory of umbral calculus based on relatively modern ideas of linear functions and linear operators. Since then, umbral calculus has been used in the study of special functions in various fields. In this article, we derive some new and interesting identities related to degenerate derangement polynomials and some special polynomials by using λ\lambda -Sheffer sequences and λ\lambda -umbral calculus, which are defined by Kim-Kim (Degenerate Sheffer sequences and λ\lambda -Sheffer sequences, J. Math. Anal. Appl. 493 (2021), 124521, 21pp).

In this paper, we consider sums related to the floor function. We improve previous results for some special arithmetic functions considered by O. Bordellès [On certain sums of number theory, Int. J. Number Theory, 18(9):2053– 2074, 2022], J. Stucky [The fractional sum of small arithmetic functions, J. Number Theory, 238:731–739, 2022], and J. Wu [Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski, Period. Math. Hung., 80(1):95–102, 2020]. It is worth emphasizing that we use much simpler methods to give much better results than previous. [ABSTRACT FROM AUTHOR]

We consider k-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number α > 1 of finite type τ < ∞ and any constant ε > 0, we can show that where Qk is the set of positive k-free integers and the implied constant depends only on α, ε, k and β. This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type. [ABSTRACT FROM AUTHOR]

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [ 1 , x ] . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized dual hyperbolic Jacobsthal numbers. As special cases, we deal with dual hyperbolic Jacobsthal and dual hyperbolic Jacobsthal-Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences. [ABSTRACT FROM AUTHOR]