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In this paper we study, in the context of weighted function spaces of continuous functions, a class of integral operators of probabilistic type, which are constructed by means of the Weibull distribution, and a suitable modification that preserves linear functions. We study the approximation properties of those two sequences of operators and we present estimates of the rate of convergence, as well as some asymptotic formulae. In particular, we prove that the proposed modification has a better rate of convergence. Finally, we show how the modified Weibull operators can be fruitfully employed to approximate suitable positive C 0 -semigroups. [ABSTRACT FROM AUTHOR]

Let $p\le 23$ be a prime and $a_p(n)$ count the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula for $\log a_p(n)$. Our results partially extend the work of Mauth, who proved the asymptotic formula for $\log a_2(n)$ conjectured by Banerjee et al.

At present, the results of the study of boundary value problems for the two-dimensional Helmholtz equation with one and two singular coefficients are known. In the presence of two positive singular coefficients in the two-dimensional Helmholtz equation, explicit solutions of the Dirichlet, Neumann and Dirichlet-Neumann problems in a quarter plane are expressed through a confluent hypergeometric function of two variables. The established properties of the confluent hypergeometric function of two variables allow us to prove the theorem of uniqueness and existence of a solution to the problems posed.In this paper, we study the Dirichlet, Neumann, and Dirichlet-Neumann problems for the three-dimensional Helmholtz equation at zero values of singular coefficients in an octant, a quarter of space, and a half-space. Uniqueness and existence theorems are proved under certain restrictions on the data. The uniqueness of solutions of which is proved using the extremum principle for elliptic equations. Using the known fundamental (singular) solution of the Helmholtz equation, solutions to the problems under study are written out in explicit forms.

The main purpose of this article is to study the problem of calculating the 2mth power mean of the generalized quadratic Gauss sums, and using the analytic method and an interesting combinatorial identity to give a sharp asymptotic formula for the 2mth power mean. Thus, a new simple proof of this existing result [N. Bag, A. Rojas-León, and W. P. Zhang, On some conjectures on generalized quadratic Gauss sums and related problems, Finite Fields Appl. 86 (2023), 10213] is provided.

Let f be any arithmetic function and define S f (x) := ∑ n ≤ x f ([ x / n ]) . If the function f is small, namely, f(n) ≪ nε, then the error term Ef(x) in the asymptotic formula of Sf(x) has the form O(x1/2+ε). In this paper, we shall study the mean square of Ef(x) and establish some new results of Ef(x) for some special functions. [ABSTRACT FROM AUTHOR]

We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l'Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l'Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus. [ABSTRACT FROM AUTHOR]

In 2007, Andrews introduced Durfee symbols and k -marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$ , and proved that the $2k$ th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$ -marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$ , $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$ -marked odd Durfee symbols of n with the i th odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$ -marked Durfee symbols of n with the i th odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright's variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$. [ABSTRACT FROM AUTHOR]

Let d (1) (n) be the n -th coefficient of the Dirichlet series (ζ ′ (s)) 2 = ∑ n = 1 ∞ d (1) (n) n − s in ℜ s > 1 , and Δ (1) (x) be the error term of ∑ n ≤ x d (1) (n). In this paper, we will study the fifth-power moment of Δ (1) (x). [ABSTRACT FROM AUTHOR]

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let σ o mex (n) (resp., σ e mex (n) ) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we find Hardy-Ramanujan type asymptotic formulae for both σ o mex (n) and σ e mex (n) . We also prove some infinite families of congruences for σ o mex (n) and σ e mex (n) modulo 4 and 8 [ABSTRACT FROM AUTHOR]

Piezoelectric thin films have been widely used in electro-acoustic devices including ones measuring Rayleigh wave speed. The main objective of this work is to find an explicit formula of Rayleigh speed taking into account the effect of the thin piezoelectric film. First the problem of Rayleigh surface waves propagating along an orthotropic substrate coated by a thin piezoelectric film is considered using Stroh's formalism approach. Then, the exact secular equation of Rayleigh wave is obtained in implicit form. In case when the thickness of the thin layer is much smaller than the considering wavelength of Rayleigh waves, an asymptotic formula of Rayleigh speed is derived from the asymptotic form of the exact secular equation. The obtained asymptotic formula is simple enough and could be used in practice when Rayleigh speed is measured by using piezoelectric device. Numerical illustrations are carried out to show the validity of the obtained approximate formulas and its potential application in inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of Durrmeyer variant of generalized Ismail operators based on Sheffer polynomials. These type of polynomials are a generalization of Appell polynomials and a special case of Boas-Buck-type polynomials. We study the convergence of these operators with the help of universal Korovkin type theorem, modulus of continuity, Peetre's $ K $-functional and the class of Lipschitz type functions. Furthermore, We extend this study to include Voronovskaja-type asymptotic results, quantitative-Voronovskaja and Gr$ \ddot{u} $ss-Voronovskaja type theorems and the convergence rate of these operators for functions in a polynomial weighted space by means of the weighted modulus of continuity. [ABSTRACT FROM AUTHOR]

We study a parametric eigenvalue problem in an infinite-dimensional Hilbert space arising in the mechanics of loaded thin-walled structures. Asymptotic properties of solutions depending on the loading parameters are established. The initial infinite-dimensional problem is approximated in a finite-dimensional subspace. Theoretical error estimates of approximate solutions are obtained. Effective numerical methods for calculating the main resonance frequency and the corresponding resonance form of vibrations based on the asymptotic formulas are proposed. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to study higher order moments of Kummer sums weighted by L-functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002). Also the results in this article give analogous results of Kummer's conjecture (1846). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is utilizing the analytic methods to study Hybrid mean value including the character, the general trigonometric sums, and the general quadratic Gauss sums, then obtain several mean value asymptotic formulae. [ABSTRACT FROM AUTHOR]

In this article, we construct a generalized Kantorovich operator based on Hermite polynomials of three variable to approximate integrable functions. For the operator, we derive the moment generating function (m.g.f.) and a few direct findings associated with m.g.f. We continue by examining the quantitative asymptotic formula and convergence in relation to the modulus of continuity and pace at which our operator converges has shown graphically. Then we capture new discrete type operator based on composition of operators and studied its local convergence and a Voronovskaja theorem in weighted space and some direct results. [ABSTRACT FROM AUTHOR]

For an odd prime $ p $ and a positive integer $ \alpha $, let $ g $ be of multiplicative order $ \tau $ modulo $ q $ and $ q = p^{\alpha} $. Denote by $ N(h, g, q) $ the number of $ a $ such that $ h\nmid (a+(g^{a})_{q}) $ for any $ 1\leq a\leq \tau $ and a fixed integer $ h\geq 2 $ with $ (h, q) = 1 $. The main purpose of this paper is to give a sharp asymptotic formula for $ N(k, h, g, q) = \mathop{\sum\limits_{\begin{smallmatrix} a = 1\\ h\nmid (a+(g^{a})_{q}) \end{smallmatrix}}^{\tau}}\left|a-(g^{a})_{q}\right| ^{2k} $ where $ k $ is any nonnegative integer and $ (a)_{q} $ denotes the smallest positive residue of $ a $ modulo $ q $. In addition, we know that $ N(h, g, q) = N(0, h, g, q) $.

The extreme narrow escape theory describes the statistical properties of the fastest among many identical stochastic particles to escape from a narrow window. We study here the escape time of the fastest particle when a killing term is added inside a one-dimensional interval. Killing represents a degradation that leads to removal of the moving particles with a given probability. Using the time dependent flux for the solution of the diffusion equation, we compute asymptotically the mean time for the fastest to escape alive. We use the present theory to study the role of several killing distributions on the mean extreme escape time for the fastest and compare the results with Brownian simulations. Finally, we discuss some applications to calcium dynamics in neuronal cells. [ABSTRACT FROM AUTHOR]

We study asymptotic behavior of solutions of nonoscillatory second‐order half‐linear differential equations. We give (in some sense optimal) conditions that guarantee generalized regular variation of all solutions, where no sign condition on the potential is assumed. For all of these solutions, we establish precise asymptotic formulas, where positive as well as negative potential is considered. We examine, as consequences, also equations with regularly varying coefficients, or with the coefficients viewed as perturbations of exponentials, or the equations under certain critical (double roots) settings. We make also asymptotic analysis of Poincaré–Perron solutions. Many of our results are new even in the linear case. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

Let h, b, c, and q ≥ 3 be integers, and let a ¯ satisfy the equation a a ¯ ≡ 1 mod q. The general mth Cochrane sum is defined as C m h , c , q = ∑ a = 1 ′ q c a ¯ / q ha m / q. The purpose of this paper is to study the hybrid mean value involving hyper-Kloosterman sums and the mth Cochrane sum ∑ x = 1 ′ q I m + 1 , h ; q C m h , c , q by applying the properties of Gauss sums, primitive characters, and a mean value theorem of Dirichlet L-functions. Similarly, we also have the hybrid mean value involving mth Cochrane sum and the general Kloosterman sum defined by Ye [Y.B. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86(3):255–267, 1998] as K m b ; c ; q : = ∑ x = 1 ′ q e bx m + c x ¯ / q. For m = 1, we obtain a better asymptotic formula than that of Zhang in [W.P. Zhang, On a Cochrane sum and its hybrid mean value formula, J. Math. Anal. Appl., 267(1):89–96, 2002]. [ABSTRACT FROM AUTHOR]

Let u = (u n) n ≥ 0 be a Lucas sequence, that is, a sequence of integers satisfying u 0 = 0 , u 1 = 1 , and u n = a 1 u n - 1 + a 2 u n - 2 for every integer n ≥ 2 , where a 1 and a 2 are fixed nonzero integers. For each prime number p with p ∤ 2 a 2 D u , where D u : = a 1 2 + 4 a 2 , let ρ u (p) be the rank of appearance of p in u , that is, the smallest positive integer k such that p ∣ u k . It is well known that ρ u (p) exists and that p ≡ (D u ∣ p) (mod ρ u (p)) , where (D u ∣ p) is the Legendre symbol. Define the index of appearance of p in u as ι u (p) : = p - (D u ∣ p) / ρ u (p) . For each positive integer t and for every x > 0 , let P u (t , x) be the set of prime numbers p such that p ≤ x , p ∤ 2 a 2 D u , and ι u (p) = t . Under the Generalized Riemann Hypothesis, and under some mild assumptions on u , we prove that # P u (t , x) = A F u (t) G u (t) x log x + O u x (log x) 2 + x log log (3 x) φ (t) (log x) 2 , for all positive integers t and for all x > t 3 , where A is the Artin constant, F u (·) is a multiplicative function, and G u (·) is a periodic function (both these functions are effectively computable in terms of u ). Furthermore, we provide some explicit examples and numerical data. [ABSTRACT FROM AUTHOR]

Let F q be the finite field with q elements. We consider the number of pairs (f 1 , f 2) with f 1 f 2 being a cube, where f i ∈ F q [ T ] and deg ⁡ f i = n for i = 1 , 2. By contour integration, we obtain an asymptotic formula in terms of either n or q. This is a polynomial analogue of a result for integers due to de la Bretèche, Kurlberg and Shparlinski [2]. [ABSTRACT FROM AUTHOR]

For an odd prime p and a positive integer α , let g be of multiplicative order τ modulo q and q = p α . Denote by N (h , g , q) the number of a such that h ∤ (a + (g a ) q) for any 1 ≤ a ≤ τ and a fixed integer h ≥ 2 with (h , q) = 1. The main purpose of this paper is to give a sharp asymptotic formula for N (k , h , g , q) = ∑ a = 1 h ∤ (a + (g a ) q) τ ⁡ | a − (g a ) q | 2 k where k is any nonnegative integer and (a) q denotes the smallest positive residue of a modulo q. In addition, we know that N (h , g , q) = N (0 , h , g , q). [ABSTRACT FROM AUTHOR]

While there are more partitions of n having more odd parts than even parts compared to partitions having more even parts than odd parts, we show that if there is a limit on the number of appearances of the part of size 1, there will eventually be more partitions of n with more even parts than odd parts compared to partitions with more odd parts than even parts. To demonstrate this, we obtain the asymptotic formulas for related partition functions via a Tauberian theorem, q-series manipulations, and partition combinatorics. [ABSTRACT FROM AUTHOR]

Consider a nonstandard continuous-time bidimensional risk model with constant force of interest, in which the two classes of claims with subexponential distributions satisfy a general dependence structure and each pair of the claim-inter-arrival times is arbitrarily dependent. Under some mild conditions, we achieve a locally uniform approximation of the finite-time ruin probability for all time horizon within a finite interval. If we further assume that each pair of the claim-inter-arrival times is negative quadrant dependent and the two classes of claims are consistently-varying-tailed, it shows that the above obtained approximation is also globally uniform for all time horizon within an infinite interval. [ABSTRACT FROM AUTHOR]

The main purpose of this article is using the analytic methods and properties of classical Gauss sums to study the calculating problem of fourth power mean values of one kind special Kloosterman's sum, and give a sharp asymptotic formula for it. At the same time, the paper also provides a new and effective method for the study of related power mean value problems.

In the present article, we introduce a new function $ \zeta $ in the modified Bernstein-Kantorovich operators defined by Özarslan and Duman (http://dx.doi.org/10.1080/01630563.2015.1079219) and properties of the function $ \zeta $ is define as: this is an infinitely differentiable function in the closed interval $ [0, 1] \; \text{such that}\; \zeta(0) = 0, \; \zeta(1) = 1 $ and $ \zeta'(\varkappa) $. We discussed an approximation properties with the help of the Bohman-Korovkin's type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions. Furthermore, we study the rate of convergence by means of derivatives of bounded variation and an approximation theorem by using of the Bohman-Korovkin's type theorem in $ \mathcal{A}- $Statistical convergence.In lastly, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATHEMATICA compiler and show that a careful choice of the function $ \zeta(\varkappa) $ for the both operators and we see that the our operator gives the better approximation results as compare to the modified Bernstein-Kantorovich operators defined by Özarslan (doi.org/10.1080/01630563.2015.1079219). [ABSTRACT FROM AUTHOR]

We consider sequences of orthogonal polynomials with varying weights, i.e., depending on the number of the polynomial. We obtain extensions of applicability classes of well-known asymptotic formulas for large numbers. [ABSTRACT FROM AUTHOR]

In [6, 7], it was shown that mixed problems can be both ill-posed for Petrovsky well-conditioned equations and well-posed for ill-conditioned equations. In the present paper we study the existence and uniqueness of the solution of a mixed problem for a class of equations with complex-valued coefficients that behave as parabolic ones, despite the fact that they can change over "time" from parabolic type to Schrödinger type, or even to antiparabolic type. [ABSTRACT FROM AUTHOR]

In this paper we study the distribution of square-free positive integers of the form and . We establish asymptotic formulas for the number of triples of positive integers such that is square-free and such that is square-free. [ABSTRACT FROM AUTHOR]

The properties of the hybrid mean value of Smarandache kn-digital series with Smarandache LCM function SL (n) are studied using elementary and combinational methods, and one sharper asymptotic formulae for it are presented．Also the main purpose of this paper is using the elementary method to study the convergent properties of the infinite series involving the Smarandache kn-digital subsequence f (m, k). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods and the number of the solutions of some congruence equations to study the calculating problem of the fourth power mean of a new two-term exponential sums, and give an exact calculating formula and a sharp asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to carry out a detailed study of a sequence of positive linear operators acting on continuous function spaces on an arbitrary real interval and constructed by means of (Borel) integrated means with respect to two families of probability Borel measures on the underlying interval and a positive real parameter. The study is mainly focused on their approximation properties in weighted spaces of continuous functions with respect to wide classes of weights. Pointwise estimates as well as weighted norm estimates are also established. In the final section a weighted asymptotic formula is obtained. [ABSTRACT FROM AUTHOR]

We investigate the triple correlation sums of coefficients of $ \theta $-series and prove an asymptotic formula with power-saving error term. As a result, we present that this type of sum is non-trivial in the regime $ H\ge X^{2/3+\varepsilon} $.

The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c > 1 $, and $ \beta $ are real numbers. The focus of the study is on solving equations of the form $ \left\lfloor \alpha n^c +\beta\right\rfloor = s m^k $, where $ m $ and $ n $ are positive integers, $ 1 \leq n \leq N $, and $ s $ is an integer. Bounds for the solutions are obtained for different values of the exponent $ k $, and an average bound is derived over $ k $-free numbers $ s $ in a given interval.

The main aim of this article is using the elementary method and the number of the solutions of some congruence equations modulo an odd prime $ p $, to study the calculating problem of the sixth power mean of one kind generalized two-term exponential sums, and give a sharp asymptotic formula for it.

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (nonmonotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude. [ABSTRACT FROM AUTHOR]

The main purpose of this article is using the analytic methods and properties of classical Gauss sums to study the calculating problem of fourth power mean values of one kind special Kloosterman's sum, and give a sharp asymptotic formula for it. At the same time, the paper also provides a new and effective method for the study of related power mean value problems. [ABSTRACT FROM AUTHOR]

Let k be a fixed integer. We study the asymptotic formula of R(H,r,k), which is the number of positive integer solutions 1 ⩽ x, y, z ⩽ H such that the polynomial x2 + y2 + z2 + k is r-free. We obtained the asymptotic formula of R(H, r, k) for all r ⩾ 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = ckH3 + O(H9/4+ε), where ck > 0 is a constant depending on k. This improves upon the error term O(H7/3+ε) obtained by G.-L. Zhou, Y. Ding (2022). [ABSTRACT FROM AUTHOR]

In this paper, the four-index generalization of the classical Le Roy function is considered on a wider set of parameters and its order and type are given. Letting one of the parameters take non-negative integer values, a family of functions with such a type of index is constructed. The behaviour of these functions is studied in the complex plane  C  and in different domains thereof. First, several inequalities are obtained in  C , and then they are modified on its compact subsets as well. Moreover, an asymptotic formula is proved for 'large' values of the indices of these functions. Additionally, the multi-index analogue of the abovementioned four-index Le Roy type function is considered and its basic properties are obtained. Finally, several special cases of the two functions under consideration are discussed. [ABSTRACT FROM AUTHOR]

The problem of evaluating square-free values of polynomials is a classical problem that has attracted many authors, including Eetermann, Carlitz, Tolev and Zhou. Recently, for 1 ≤ x , y ≤ H , Dimitrov established an asymptotic formula for the number of the square-free values attained by the polynomial f x , y = x 2 + y 2 + 1 x 2 + y 2 + 2 . Motived by the work of Dimitrov, in this paper, we give an asymptotic formula for the consecutive square-free numbers x 1 2 + ⋯ + x k 2 + 1 , x 1 2 + ⋯ + x k 2 + 2 . [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. We consider sums of type ∑ n 1 ⋯ n k ≤ x F (n 1 , ... , n k) , taken over the hyperbolic region { (n 1 , ... , n k) ∈ N k : n 1 ⋯ n k ≤ x } , where F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ∑ n 1 ⋯ n k ≤ x f ((n 1 , ... , n k)) and ∑ n 1 ⋯ n k ≤ x f ([ n 1 , ... , n k ]) , involving the GCD and LCM of the integers n 1 , ... , n k , where f : N → C belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k = 2 . [ABSTRACT FROM AUTHOR]