scholarly journals Repeated sums and binomial coefficients

2021 ◽  
Vol 4 (2) ◽  
pp. 30-47
Author(s):  
Roudy El Haddad ◽  
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify some particular sums such as the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive formulae for simplifying general <i> repeated sums</i> as well as a variant containing binomial coefficients. Additionally, we study the \(m\)-th difference of a sequence and show how sequences whose \(m\)-th difference is constant can be related to binomial coefficients.

scholarly journals Analogues of the binomial coefficient theorems of Gauss and Jacobi

2016 ◽  
Vol 12 (08) ◽  
pp. 2125-2145
Author(s):  
Abdullah Al-Shaghay ◽  
Karl Dilcher
Keyword(s):  
Gamma Function ◽  
Simple Form ◽  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

Preserving log-concavity for p,q-binomial coefficient

2019 ◽  
Vol 11 (02) ◽  
pp. 1950017
Author(s):  
Moussa Ahmia ◽  
Hacène Belbachir
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Transformations ◽  
Pascal Triangle

We study the log-concavity of a sequence of [Formula: see text]-binomial coefficients located on a ray of the [Formula: see text]-Pascal triangle for certain directions, and we establish the preserving log-concavity of linear transformations associated to [Formula: see text]-Pascal triangle.

scholarly journals On Binomial Coefficient Residues

1957 ◽  
Vol 9 ◽  
pp. 363-370 ◽  
Author(s):  
J . B. Roberts
Keyword(s):  
Prime Number ◽  
Difference Equations ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Difference Equations ◽  
Constant Coefficients

The number of binomial coefficients , which are congruent to j , 0 ≤ j ≤ p − 1, modulo the prime number p is denoted by θj(n). In this paper we give systems of simultaneous linear difference equations with constant coefficients whose solutions would yield the quantities θj(n) explicitly.

Stop-sign theorems and binomial coefficients

2010 ◽  
Vol 94 (530) ◽  
pp. 247-261 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Quality Of Life ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Original Form ◽  
Pascal Triangle ◽  
Stop Sign

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

scholarly journals INTEGRAL REPRESENTATIONS AND INEQUALITIES OF EXTENDED CENTRAL BINOMIAL COEFFICIENTS

2021 ◽  
Author(s):  
Chunfu Wei
Keyword(s):  
Binomial Coefficient ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Exponential Functions ◽  
Central Binomial Coefficient

In the paper, the author presents three integral representations of extended central binomial coefficient, proves decreasing and increasing properties of two power-exponential functions involving extended (central) binomial coefficients, derives several double inequalities for bounding extended (central) binomial coefficient, and compares with known results.

Binomial Coefficient Formulas by General Reasoning

1968 ◽  
Vol 61 (4) ◽  
pp. 399-402
Author(s):  
Jack M. Elkin
Keyword(s):  
Binomial Coefficient ◽  
Algebraic Manipulation ◽  
Binomial Coefficients ◽  
Binomial Theorem ◽  
Relative Ease ◽  
Methods Of Analysis ◽  
Summation Of Series ◽  
Mathematical Formulas ◽  
High Degree ◽  
Pascal's Triangle

The binomial coefficients are an almost endless source of formulas for the summation of series. A reference to the “Problems for Solution” pages of the American Mathematical Monthly or to an advanced collection of mathematical formulas will convince anyone who has not yet discovered this for himself. Some of these series summations can be derived with relative ease with the help of the binomial theorem or Pascal's triangle; many require a high degree of virtuosity in algebraic manipulation and, often, advanced methods of analysis. A number of them can be obtained simply by reasoning logically about the meaning of certain combinatorial expressions, with recourse to only a minimum of algebra or to none at all. These, naturally, have a special appeal of their own, and it is the purpose of this article to illustrate several such derivations.

Tribinomial coefficients and Catalan numbers

2009 ◽  
Vol 93 (528) ◽  
pp. 449-455 ◽  
Author(s):  
Thomas Koshy ◽  
Mohammad Salmassi
Keyword(s):  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Triangular Numbers ◽  
Interesting Family ◽  
Positive Integers

The concept of the ordinary binomial coefficientcan be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

scholarly journals Evaluation of a Special Hankel Determinant of Binomial Coefficients

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Ömer Eugeciouglu ◽  
Timothy Redmond ◽  
Charles Ryavec
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Constant Term ◽  
Product Evaluation ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Operator Technique ◽  
Hankel Determinant ◽  
International Audience ◽  
Convolution Equations

International audience This paper makes use of the recently introduced technique of $\gamma$-operators to evaluate the Hankel determinant with binomial coefficient entries $a_k = (3 k)! / (2k)! k!$. We actually evaluate the determinant of a class of polynomials $a_k(x)$ having this binomial coefficient as constant term. The evaluation in the polynomial case is as an almost product, i.e. as a sum of a small number of products. The $\gamma$-operator technique to find the explicit form of the almost product relies on differential-convolution equations and establishes a second order differential equation for the determinant. In addition to $x=0$, product form evaluations for $x = \frac{3}{5}, \frac{3}{4}, \frac{3}{2}, 3$ are also presented. At $x=1$, we obtain another almost product evaluation for the Hankel determinant with $a_k = ( 3 k+1) ! / (2k+1)!k!$.

scholarly journals Asymptotic approximation of central binomial coefficients with rigorous error bounds

2021 ◽  
Vol 5 (1) ◽  
pp. 380-386
Author(s):  
Richard P. Brent ◽  
Keyword(s):  
Theta Function ◽  
Error Bounds ◽  
Asymptotic Approximation ◽  
Asymptotic Series ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Rigorous Error Bounds ◽  
Historical Remarks ◽  
Rigorous Error ◽  
Central Binomial Coefficient

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.

scholarly journals Central binomial coefficients divisible by or coprime to their indices

2018 ◽  
Vol 14 (04) ◽  
pp. 1135-1141 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Upper Density ◽  
Positive Integers ◽  
Central Binomial Coefficient

Let [Formula: see text] be the set of all positive integers [Formula: see text] such that [Formula: see text] divides the central binomial coefficient [Formula: see text]. Pomerance proved that the upper density of [Formula: see text] is at most [Formula: see text]. We improve this bound to [Formula: see text]. Moreover, let [Formula: see text] be the set of all positive integers [Formula: see text] such that [Formula: see text] and [Formula: see text] are relatively prime. We show that [Formula: see text] for all [Formula: see text].

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