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Several sharp approximations of the generalized-binomial-coefficient function having real arguments are presented on the basis of Stirling's approximation formula for Γ function. [ABSTRACT FROM AUTHOR]

In this paper we consider a number triangle known as the Rascal triangle and place it within an infinite sequence of number triangles we call Iterated Rascal triangles. Initially, the name "Rascal triangle" was a clever play of words. However, we show a surprising and direct connection between Iterated Rascal triangles and Pascal's triangle. We prove an unexpected and elegant recursive formula generates the entries in any Iterated Rascal triangle. We establish formulas, generating functions, and combinatorial identities for them. From these principals we prove a version of Lucas' Theorem for the Iterated Rascal triangles. Moreover, we consider an identity from the Rascal triangle that has a well-known analogous identity in Pascal's triangle and extend it to all Iterated Rascal triangles. Although many can be made, we conclude with one conjecture. [ABSTRACT FROM AUTHOR]

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics. [ABSTRACT FROM AUTHOR]

Three classes of infinite series containing binomial coefficient (3nn), harmonic-like numbers, and an independent variable “y” are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert’s series and manipulating the cubic transformations for the 3F2-series through the “coefficient extraction” method. [ABSTRACT FROM AUTHOR]

Three classes of infinite series containing binomial coefficient $ \binom{3n}{n} $, harmonic-like numbers, and an independent variable "$ y $" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the $ _3{F_2} $-series through the "coefficient extraction" method.

Five classes of multiple sums about circular products of binomial coefficients are investigated. Their generating functions are explicitly expressed as rational functions, with coefficients being Fibonacci and Lucas numbers. This is fulfilled by a novel approach called "recursive construction", which also permits us to establish, for the circular sums, both generating functions and recurrence relations through the Lagrange expansion formula. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

We give an analogue of Chapman’s extension to Sun’s symmetric identities for the discrete Appell sequences. [ABSTRACT FROM AUTHOR]

In this paper, we extend tools developed in [9] to study Euler T-type sums involving odd harmonic numbers and binomial coefficients. In particular, we will prove that two kinds of Euler T-type sums can be expressed in terms of log(2), zeta values, double T-values, (odd) harmonic numbers and double T-sums. [ABSTRACT FROM AUTHOR]

We present some new linear, quadratic, cubic, and quartic binomial Fibonacci, Lucas, and Fibonacci--Lucas summation identities. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals.

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

By employing one of the cubic transformations (due to W.N. Bailey (1928)) for the 3F2(x)-series, we examine a class of 3F2(4)-series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Min matrices [Formula: see text] and Max matrices [Formula: see text] whose entries are [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] and [Formula: see text]. By utilizing combinatorial identities, we derive the characteristic polynomials of M and U. Additionally, relationship between generalized Fibonacci numbers and the characteristic polynomials of these matrices is established. Finally, some examples are given to illustrate our results.

Five classes of multiple sums about circular products of binomial coefficients are investigated. Their generating functions are explicitly expressed as rational functions, with coefficients being Fibonacci and Lucas numbers. This is fulfilled by a novel approach called “recursive construction”, which also permits us to establish, for the circular sums, both generating functions and recurrence relations through the Lagrange expansion formula.

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]

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]

Pascal was a great mathematician and scientist, who contributed to many fields in mathematics and science. When he was 19 years old, he developed the first calculator, and together with Fermat he was the founder of probability theory. He investigated the properties of a triangle of numbers, which is named today "the Pascal triangle" and developed the method of proving theorems by mathematical induction. Pascal also investigated the properties of the cycloid, and he conducted the physical experiment that proved the existence of the void. After a spiritual experience at the age of 32, Pascal left mathematics and science altogether and dedicated himself to investigating and writing about religion. This paper suggests the new language of Soft logic, which is based on the extension of the number 0 to the zero axis. We conclude by an example of the extension of the Pascal Triangle with Soft numbers. Also, we discuss the possibility to develop a new model of computation. [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]

We provide an upper bound on binomial coefficients that holds over the entire parameter range an whose form repeats the form of the de Moivre – Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the number of continuations of a given Boolean function to bent functions, investigate dependencies into the Walsh – Hadamard spectra, obtain restrictions on the number of representations as sums of squares of integers bounded in magnitude.

By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz ['The characteristic polynomial of a certain matrix of binomial coefficients', Fibonacci Quart. 3 (2) (1965), 81–89]. [ABSTRACT FROM AUTHOR]

By means of the generating function approach, four classes of triple sums involving circular products of binomial coefficients are investigated. Recurrence relations and rational generating functions are established. [ABSTRACT FROM AUTHOR]

We evaluate, in closed forms, twelve infinite series that contain a free variable x and binomial coefficients ( 3 n + ϵ n) in the denominators. Several summation formulae resembling Gosper's sum are derived as consequences. [ABSTRACT FROM AUTHOR]

Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.

We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.

In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.

Let (in) = n!/i!(n - i)! (n, i ∈ N*, and n ≥ i) be a binomial coefficient. Supposing that l is a nonnegative integer and Ll is the l-th Lucas number, furthermore, when n is a fixed positive integer and k is a fixed nonnegative integer, the convolution of the sequence ... and ... is l(k, p, n) = ... When p=4, the identity l(k, 4, n) = 3n L4k+2n + 4( - 1)k+nL2k+n + 3 ×2n+1 is proved by elementary number theory method. [ABSTRACT FROM AUTHOR]

The Sandor-Smarandache function, SS(n), is a recently introduced Smarandache-type arithmetic function, which involves binomial coefficients. It is known that SS(n) does not possess many of the common properties of the classical arithmetic functions of the theory of numbers. Sandor gave the expression of SS(n) when n (≥ 3) is an odd integer. It is found that SS(n) has a simple form when n is even and not divisible by 3. In the previous papers, some closed-form expressions of SS(n) have been derived for some particular cases of n. This paper continues to find more forms of SS(n), starting from the function SS(24m). Particular attention is given to finding necessary and sufficient conditions such that SS(n) = n-5 and SS(n) = n-6. Based on the properties of SS(n), some interesting Diophantine equations have been studied. The study reveals that the form of SS(n) depends on the prime factors of the integer n in the natural order of the primes. [ABSTRACT FROM AUTHOR]

Nearest-neighbor-spacing distributions of people sitting along the west bank of the Kamogawa River flowing through the central part of Kyoto City are derived, based on a dual partitioning process with a fixed ratio. The theoretical asymmetric bell-shaped distribution is compared with previous measurements. This was performed as an advanced study of high school students that does not follow the curriculum guidelines of Japanese high schools. [ABSTRACT FROM AUTHOR]

In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory. [ABSTRACT FROM AUTHOR]

Self-dual codes over the ring Z 4 are related to combinatorial designs and unimodular lattices. First, we discuss briefly how to construct self-dual cyclic codes over Z 4 of arbitrary even length. Then we focus on solving one key problem of this subject: for any positive integers k and m such that m is even, we give a direct and effective method to construct all distinct Hermitian self-dual cyclic codes of length 2k over the Galois ring GR(4,m). This then allows us to provide explicit expressions to accurately represent all these Hermitian self-dual cyclic codes in terms of binomial coefficients. In particular, several numerical examples are presented to illustrate our applications. [ABSTRACT FROM AUTHOR]

Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

Let t ∈ N + be given. In this article, we are interested in characterizing those d ∈ N + such that the congruence 1 t ∑ s = 0 t - 1 n + d ζ t s d - 1 ≡ n d - 1 (mod d) is true for each n ∈ Z . In particular, assuming that d has a prime divisor greater than t, we show that the above congruence holds for each n ∈ Z if and only if d = p r , where p is a prime number greater than t and r ∈ { 1 , ... , t } . [ABSTRACT FROM AUTHOR]

In the paper, with the help of Kazarinoff's integral representation for the ratio of two gamma functions, with the aid the duplication formula of the digamma function, by virtue of integral representations of polygamma functions, and in the light of the L'Hôpital type monotonicity rule, the author describes extended binomial coefficients in terms of the gamma functions and the falling factorials, collects three integral representations of central binomial coefficients, establishes three integral representations of extended central binomial coefficients, proves decreasing and increasing properties of two power‐exponential functions involving extended (central) binomial coefficients, and derives several double and sharp inequalities for bounding extended (central) binomial coefficients, and compares these inequalities with known ones. [ABSTRACT FROM AUTHOR]

We use a payment pattern of the type { 1 k , 2 k , 3 k , ... } to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of an n-year polynomial annuity, the present value of an m-monthly n-year polynomial annuity, and the present value of an n-year continuous polynomial annuity. We also use the idea to extend the annuities to payment patterns derived from analytic functions, as well as to payment patterns of the type { 1 r , 2 r , 3 r , ... } , with r being an arbitrary real number. In the process, we develop possible approximations to k ! and for the gamma function evaluated at real numbers. [ABSTRACT FROM AUTHOR]

Copyright of Journal de Ciencia e Ingeniería is the property of Corporacion Universitaria Autonoma del Cauca 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.)

Abstract Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions.

We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the nth word is uniformly distributed over the set of words of length 2n in which n letters are a and n letters are b: at each step an a and a b are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain and thereby delineate all the ways in which the Markov chain can be conditioned to behave at large times. Writing N(u) for the number of letters a (equivalently, b) in the finite word u, we show that a sequence (un ) n∈ℕ of finite words converges to a point in the boundary if, for an arbitrary word ν, there is convergence as n tends to infinity of the probability that the selection of N(ν) letters a and N(ν) letters b uniformly at random from un and maintaining their relative order results in ν. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set {a1, b1, a2, b2, …} that have distributions which are separately invariant under finite permutations of the indices of the a's and those of the b's. We establish a further bijective correspondence between the set of such random total orders and the set of pairs (μ, ν) of diffuse probability measures on [0,1] such that ½(μ + ν) is Lebesgue measure: the restriction of the random total order to {a1, b1,…, an, bn } is obtained by taking X1,…, Xn (resp. Y1,… ,Yn ) i.i.d. with common distribution μ (resp. ν), letting (Z1,…, Z2n) be {X1, Y1,…, Xn , Yn } in increasing order, and declaring that the kth smallest element in the restricted total order is ai (resp. bj ) if Zk = Xi (resp. Zk = Yj ).

A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms. [ABSTRACT FROM AUTHOR]

In this paper, we establish a non-trivial upper bound for the exponential sum ∑ y < n < y ′ Λ (n) e (x / n) when y < y ′ ≤ 2 y. [ABSTRACT FROM AUTHOR]

We prove a general combinatorial formula involving the reciprocals of the binomial coefficients and the partial sum of an arbitrary sequence. Applying this formula we offer many combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbers; some of which have been considered previously and the others are new. [ABSTRACT FROM AUTHOR]

The Sandor-Smarandache function, due to Sandor, has drawn the attention of the researchers soon after its introduction. The new Smarandache-type arithmetic function, denoted by SS (n), involves binomial coefficients. Sandor found SS(n) when n ( ³ 3) is an odd integer. It has been shown that the function has a simple form even when n is even and not divisible by 3. In earlier papers, some closed-form expressions of SS(n) have been derived for particular cases of n. Still, some unexplored results are needed to settle down the function. So, this study finds more forms of SS(n), starting from the function SS(24m). Particular attention has been focused on the functions SS(120m), SS(840m), SS(9240m) and SS(120120m). [ABSTRACT FROM AUTHOR]

Copyright of Journal Of Sichuan University (Natural Sciences Division) / Sichuan Daxue Xuebao-Ziran Kexueban is the property of Editorial Department of Journal of Sichuan University Natural Science Edition 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.)