Search

In this paper, we mainly prove the following result: For any positive integers l and n and nonnegative integer k with k ≤ n - 1 , we have (2 l - 1) ! ! ∑ m = k n - 1 (2 m + 1) 2 l - 1 m + k 2 k 2 k k 2 ≡ 0 mod n 2 n - 1 k n + k k . This confirms a conjecture of Sun and its extension by Guo. [ABSTRACT FROM AUTHOR]

The object of research in the paper is extrapolation problems based on interpolation polynomials. Polynomial-based prediction methods are well known. However, the problem is that such methods often give very large errors in practice. The permissible error of extrapolation even by one grid step is not ensured by the high accuracy of interpolation using polynomials. The paper proposes an algorithm that allows to significantly improve polynomial forecasts by optimizing the procedure for choosing the power of the polynomial, on the basis of which the forecast is built. The algorithm is based on the procedure for building all polynomial forecasts according to known data and analysis of these forecasts. In particular, the presence of monotonicity and a tendency to convergence allows determining the optimal degree of the polynomial. In the absence of monotonicity, provided that certain ratios are met, the forecast can be constructed as the arithmetic average of all polynomial forecasts. An important result is the estimation of the error of the forecasting method by averaging polynomial forecasts. The development of the algorithm became possible due to the use of a special method of constructing a one-step polynomial forecast. The method differs in that it allows to build a forecast without using the cumbersome procedure of calculating the unknown coefficients of the polynomial. The numerical results presented in the work demonstrate the effectiveness of the forecasting technique based on the average of polynomial forecasts. In particular, for the test functions, the relative error was about 2–5 %, while polynomials of different degrees in the worst case yielded more than 50 %. The obtained results can be useful for building shortterm forecasts of series of economic dynamics, forecasting the behavior of arbitrary processes with a dominant deterministic component. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q ; h) -time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator α that combines both jump operators in a convex manner. This allows us to generate a time scale α , which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale α , which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on α -time scales using the α -operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the α -version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus. [ABSTRACT FROM AUTHOR]

In this paper, we present several new congruences on the q-trinomial coefficients introduced by Andrews and Baxter [J. Statist. Phys. 47 (1987), 297–330]. A new congruence on sums of central q-binomial coefficients is also established. [ABSTRACT FROM AUTHOR]

This paper provides another proof of Fermat's theorem. As in the previous work, a geometric approach is used, namely: instead of integers a, b, c, a triangle with side lengths a, b, c is considered. To preserve the completeness of the proof of the theorem in this work, the proof is repeated for the cases of right and obtuse triangles. In this case, the Fermat equation ap+bp = cp has no solutions for any natural number p > 2 and arbitrary numbers a, b, c. When considering the case when the numbers a, b, c are sides of an acute triangle, it is proven that Fermat's equation has no solutions for any natural number p > 2 and non-zero integer numbers a, b, c. Numbers a = k, b = k+m, c = k+n, where k, m, n are natural numbers that satisfy the inequalities n > m, n < k+m, exhaust all possible variants of natural numbers a, b, c, which are the sides of the triangle. In an acute triangle, the following condition is additionally satisfied: k > n -m + √2n(n - m). To study the Fermat equation, an auxiliary function f(k,p) = kp+(k+m)p-(k+n)p, is introduced, which is a polynomial of natural degree p in the variable k. The equation f(k, p) = 0 has a single positive root for any natural p = 2. A recurrent formula connecting the functions f(k,p+1) and f(k,p) has been proven: f(k,p+1) = kf(k, p)-[n(k+n)p-m(k+m)p]. The proof of the main proposition 2 is based on considering all possible relationships between the assumed integer solution of the equation f(k, p+1) = 0 and the number (ñ -m) corresponding to this solution k. The proof was carried out using the mathematical apparatus of number theory, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat's theorem were proved. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper, using some properties of Fibonacci numbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, we establish some congruences involving central binomial coefficient modulo p and p². We also give some new identities for hyperbolic functions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a0 + a1x + a2x2 + · · · + anxn. It is shown that several sequences a give the trivial sequence b = (0, 1, 0, 1, 0, 1,...), i.e., bn = n mod 2, among them the Catalan numbers, central binomial coefficients, n! and for a fixed k. We also look at some sequences a for which b is more interesting such as an = (n + 1)k for k ≥ 3. Further, general procedures are given for constructing real sequences an for which bn is either always maximal or minimal. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums. [ABSTRACT FROM AUTHOR]

By employing the coefficient extraction method from hypergeometric series, we shall establish numerous closed form evaluations for infinite series containing central binomial coefficients and harmonic numbers, including several conjectured ones made by Z.-W. Sun. [ABSTRACT FROM AUTHOR]

In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime p>3 and positive odd integer m, we prove that there is a p-adic integer c_m only depending on m such that \begin{equation*} \sum _{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left (\frac {p}{3}\right)\pmod {p^3}, \end{equation*} where A_k=\sum _{j=0}^{k}\binom {k}{j}^2\binom {k+j}{j}^2 is the Apéry number and (\frac {.}{p}) is the Legendre symbol. [ABSTRACT FROM AUTHOR]

In this paper, we established an effective formula for Dnestrovskii functions based on the binomial expansion theorem which frequently appears in a weakly relativistic loss cone distribution, relativistic, magnetized, and thermal plasmas. The obtained formulas are expressed in terms of binomial coefficients and incomplete Gamma functions. Therefore, the proposed approach is a fully useful method to evaluate Dnestrovskii functions and related physical properties of plasmas. Calculation results of the Dnestrovskii functions are compared with other numerical and theoretical models and indicated acceptable agreement. The results of the calculations demonstrate that the derived formula is valid for a wide range of parameters. [ABSTRACT FROM AUTHOR]

We discuss the probabilistic question faced by the Italian scholar Giovanni Rizzetti (1675–1751) and suggest this is a variant of what we refer to today as the " ballot sequences ", variant to which Catalan numbers offer a solution. Rizzetti's search for a solution led him to produce an iterative rule which involved the consideration of Catalan numbers. Although coming up with these were not his final goal, the rule he elaborated allowed him to carry on calculating them as long as he wanted. As such, it is possible this was the first time such numbers were determined. • Giovanni Rizzetti (1675–1751) probability scholar in the group of Jacopo Riccati. • The historical work of Igor Pak and the birth of Catalan numbers. • The denomination "Catalan numbers" (from Igor Pak). • Catalan numbers as a solution to a probability problem in a manuscript of Rizzetti. • Rizzetti's solution and the first calculation of Catalan numbers. [ABSTRACT FROM AUTHOR]

Previous studies have found a positive relationship between the youth and the educated with protest number, but the form that these protests take needs further research. We argue that students are a unique group, acting neither as an educated nor a young population, and three possible mechanisms push students toward non-violent rather than violent forms of protest. By promoting values of tolerance, higher levels of human capital, and social mobility, education serves as a factor that pacifies destructive tendencies in protest movements. At the same time, universities are a platform for cooperation, and the large amounts of free time and energy make the costs of participating in protests for students minimal compared with other groups. Using a negative binomial regression and a rare events logistic regression, we find that the proportion of students is a strong and consistently significant predictor of the number of nonviolent demonstrations. However, the share of students in the total population does not turn out to be significantly associated with violent protests/armed uprisings. [ABSTRACT FROM AUTHOR]

The presence of global spatial autocorrelation usually leads to the spurious identification of spatial hotspots and hinders the identification of local hotspots. Despite the use of statistical methods to address global spatial autocorrelation in spatial hotspot detection, accurately modeling global spatial autocorrelation structure without the stationarity assumption of spatial processes is difficult. To overcome this challenge, we fitted the global spatial autocorrelation structure from a geometric perspective and identified the optimal global spatial autocorrelation structure by analyzing the variances in spatial data. Hotspots were detected from the residuals obtained by removing the global spatial autocorrelation structure from the original dataset. We upgraded a weighted moving average method based on binomial coefficients (Yang Chizhong filtering) to fit the global spatial autocorrelation structure for field-like geographic phenomena. A variance decay indicator, based on the variance in the original and filtered data, was used to identify the optimal global spatial autocorrelation structure. Yang Chizhong filtering does not require a spatial stationarity assumption and can preserve local autocorrelation structures in the residuals as much as possible. Experimental results showed that hotspot detection methods combined with Yang Chizhong filtering can effectively reduce type-I and -II errors in the results and discover implicit and valuable urban hotspots. [ABSTRACT FROM AUTHOR]

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the fourth order and all non-central moments up to the eighth order. [ABSTRACT FROM AUTHOR]

The combinatorial nature of the trigonometric integrals (2.8) is discussed in connection to the partition of multisets with equal sums. Computational aspects are highlighted for special parameter values. [ABSTRACT FROM AUTHOR]

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. [ABSTRACT FROM AUTHOR]

We study the conjugation of overpartitions and give the generating function for the number of self-conjugate overpartitions of an integer. Following the recent introduction of over q-binomial coefficients, we obtain the over q-analogue of the Chu-Vandermonde identity. Consequently a new generating function for the number of overpartitions is proved. We also give a new over q-analogue of the Chu-Vandermonde identity. [ABSTRACT FROM AUTHOR]

An accurate approach is proposed for the temperature dependence of electron concentration and Lorenz number in the spherically symmetric zone of semiconductors. The evaluation includes more accurate analytical calculations over the study for two parameters of Fermi functions. Recently, a new analytical approach for the calculation of the two parameters of Fermi functions has been reported in terms of summations of binomial coefficients and incomplete gamma functions. The method is applied to the case of the Ge and GaAs semiconductors, which can determine the electron concentration and Lorenz number as a function of temperature variation. The results obtained by the suggested and numerical methods are satisfactory for a wide range of temperatures. The method descriptions are very well for the investigation of other thermoelectric effects over the whole temperature ranges. [ABSTRACT FROM AUTHOR]

For two matrices A and B, and large n, we show that most products of n factors of e A / n and n factors of e B / n are close to e A + B . This extends the Lie-Trotter formula. The elementary proof is based on the relation between words and lattice paths, asymptotics of binomial coefficients, and matrix inequalities. The result holds for more than two matrices. [ABSTRACT FROM AUTHOR]

Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients. Among them, three conjectured identities due to Z.-W. Sun are also confirmed. [ABSTRACT FROM AUTHOR]

We explicitly determine the Apéry limits for the sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's conjecture on the order of the recurrences for these sequences. Namely, we prove the conjectured minimal order under the assumption that such a recurrence can be obtained via creative telescoping. [ABSTRACT FROM AUTHOR]

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length p k over the Galois ring GR (p 2 , m) is the key to construct self-dual cyclic codes of length p k n over the integer residue class ring Z p 2 for any positive integer n satisfying gcd (p , n) = 1 . So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length p k over GR (p 2 , m) by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients. [ABSTRACT FROM AUTHOR]

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers. [ABSTRACT FROM AUTHOR]

In 1979, Apéry proved the irrationality of ζ (2) and ζ (3) . Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman's results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjecture of Sun[Congruences involving binomial coefficients and Apery-like numbers, Publ. Math. Debrecen 96 (2020), pp. 315–346]: Let p>3 be a prime. Then ∑ k = 0 p − 1 ( 2 k k ) 3 k + 1 (− 16) k f k ≡ (− 1) (p − 1) / 2 p + p 3 E p − 3 (mod p 4) , where f n = ∑ k = 0 n ( n k ) 3 and E n stand for the nth Franel number and nth Euler number, respectively. [ABSTRACT FROM AUTHOR]

We consider the sequence of integers whose nth term has base-p expansion given by the nth row of Pascal's triangle modulo p (where p is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a 2-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo p. This note ends with a discussion about Pascal's pyramid built with trinomial coefficients. [ABSTRACT FROM AUTHOR]

There is an asymptotic correspondence between the multiplicative relations among multinomial coefficients and the (additive) recursive property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a 1-cocycle condition in information cohomology, an algebraic invariant of presheaves of modules on certain categories of observables. Depending on the coefficients, the 1-cocycles can be information measures (Shannon entropy, Tsallis α -entropy) or generalized (Fontené-Ward) multinomial coefficients. In each case the 1-cocycle condition encodes a system of functional equations. We obtain in particular a combinatorial analogue of the "fundamental equation of information theory": a simple functional equation that uniquely characterizes the generalized binomial coefficients. The asymptotic correspondence mentioned above extends to any α -entropy and certain multinomial coefficients with compatible asymptotic behavior, shedding new light on the meaning of the chain rule and its deformations. [ABSTRACT FROM AUTHOR]

We consider the problem of partitioning of space by hyperplanes that arises in many application areas, where the number of regions the space is divided into is required to be determined, such as speech/pattern recognition, various classification problems, data analysis. We obtain some relations for the number of divisions and establish a recurrence relation for the maximum number of regions in d-dimensional Euclidean space cut by n hyperplanes. We also re-derive an explicit formula for the number of regions into which the space can be partitioned by n hyperplanes. [ABSTRACT FROM AUTHOR]

We consider Bernoulli measures \mu _p on the interval [0,1]. For the standard Lebesgue measure the digits 0 and 1 in the binary representation of real numbers appear with an equal probability 1/2. For the Bernoulli measures, the digits 0 and 1 appear with probabilities p and 1-p, respectively. We provide explicit expressions for various \mu _p-integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of \mu _p in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire functions, which admit explicit infinite product expansions and satisfy interesting properties, including connections with the Stirling numbers and the polylogarithm. [ABSTRACT FROM AUTHOR]

In this paper, we prove that all but one of the components of the transfer digraph Dm* needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width m (m ∈ N) are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, we prove that the set of vertices of each component consists of all the binary m-words for which the difference of numbers of zeros in odd and even positions is constant. [ABSTRACT FROM AUTHOR]

We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers. [ABSTRACT FROM AUTHOR]

This paper was inspired by Donald Knuth’s celebrated explanation of the remarkable connection between q-binomial coefficients and integer partitions. In the spirit of Knuth’s proof, we offer a new proof of the well-known result that a certain q-analogue of multinomial coefficients furnishes a q-counting of the set of permutations of an associated multiset of positive integers, according to the number of inversions in such arrangements. Our proof uses the fact that such q-multinomial coefficients enumerate certain classes of chains of subspaces of a finite dimensional vector space over a finite field of cardinality q. Additionally, we investigate the function that counts the number of permutations of a multiset having a fixed number of inversions. [ABSTRACT FROM AUTHOR]

Using a hypergeometric series approach, a general combinatorial identity is found in this note, and among its special cases are well-known and classical combinatorial identities due to Gould and Touchard. [ABSTRACT FROM AUTHOR]

A simple complex Lie algebra is characterized by a root system and a Chevalley basis with the integer structure constants. The well-known arbitrariness of their choice for the niltriangular subalgebra essentially affects the Lie-admissible algebra (in the sense of Albert) over a field such that . We study the uniqueness of the (nonassociative) enveloping algebras of classical types. The enumeration of ideals of the Lie algebras and for leads to the solution of some combinatorial problem listed in ACM SIGSAM Bulletin in 2001. The calculations of multiple combinatorial sums with -binomial coefficient use the integral representation method of combinatorial sums (the coefficient method). [ABSTRACT FROM AUTHOR]

In the present work we give an extension of (p,q)-analysis. As an extension of (p,q)-analysis, the (r,p,q)-analysis is introduced. We define some elementary concepts of this analysis such as (r,p,q)-numbers, (r,p,q)-derivative, (r,p,q)-exponential functions, (r,p,q)-antiderivative and (r,p,q)-integral. We obtain some properties of the polynomial (x -- a)n (r,p,q)-Taylor formula, (r,p,q)-binomial coefficients, divided differences and some relations between (r,p,q)-derivative, (r,p,q)-exponential functions, (r,p,q)-integral and finally, the fundamental theorem of (r,p,q)-analysis are examined in details. [ABSTRACT FROM AUTHOR]

For any m , n ∈ N = { 0 , 1 , 2 ... } , the truncated hypergeometric series m + 1 F m is defined by m + 1 F m [ x 0 x 1 ... x m y 1 ... y m | z ] n = ∑ k = 0 n (x 0) k (x 1) k ⋯ (x m) k (y 1) k ⋯ (y m) k ⋅ z k k ! , where (x) k = x (x + 1) ⋯ (x + k − 1) is the Pochhammer symbol. Let p be an odd prime. For α , z ∈ Z p with ⟨ − α ⟩ p ≡ 0 (mod 2) , where ⟨ x ⟩ p denotes the least nonnegative residue of x modulo p for any x ∈ Z p , we mainly prove the following congruence motivated by Orr's identity: 2 F 1 [ 1 2 α 3 2 − 1 2 α 1 | z ] p − 1 2 F 1 [ 1 2 α 1 2 − 1 2 α 1 | z ] p − 1 ≡ 3 F 2 [ α 2 − α 1 2 1 1 | z ] p − 1 (mod p 2). As a corollary, for any positive integer b with p ≡ ± 1 (mod b) and ⟨ − 1 / b ⟩ p ≡ 0 (mod 2) , we deduce that ∑ k = 0 p − 1 (b 2 k + b − 1) ( 2 k k ) 4 k ( − 1 / b k ) ( 1 / b − 1 k ) ≡ 0 (mod p 2). This confirms a conjectural congruence of the second author. [ABSTRACT FROM AUTHOR]

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular, we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0 , in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1 , the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < 1 , the formal solutions are generically 1-Gevrey. In the latter case, we give in particular an explicit example in which the formal solution is s ′ -Gevrey for no s ′ < 1 , that is exactly 1-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are 1-summable in a given direction arg (t) = θ. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results. [ABSTRACT FROM AUTHOR]

The main objective in this paper is to study on special numbers and polynomials that contain finite sums of powers of binomial coefficients. By using generating function methods, some formulas and relations related to these numbers and the Apostol-Bernoulli and Apostol-Euler numbers of negative higher order, the Bernoulli and Euler numbers, the Stirling type numbers, the combinatorial numbers, the Bell polynomials, the Fubini type polynomials, and the Legendre polynomials are presented. Moreover, asymptotic expressions of the finite sums of powers of binomial coefficients for these numbers are given. Some numeric values of these asymptotic expressions are illustrated by the tables. Finally, some inequalities for these numbers are given. [ABSTRACT FROM AUTHOR]

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that 'tweaked' the sum of the sth powers of the n-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call 'major Apéry miracles', proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a 'minor Apéry miracle' where an explicit constant, defined as an (extremely slowly converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error. [ABSTRACT FROM AUTHOR]