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.

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 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.

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals Generalized Pascal triangle for binomial coefficients of words

2016 ◽  
Vol 80 ◽  
pp. 24-47 ◽  
Author(s):  
Julien Leroy ◽  
Michel Rigo ◽  
Manon Stipulanti
Keyword(s):  
Binomial Coefficients ◽  
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.

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 Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

10.37236/5913 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Lily Li Liu ◽  
Ya-Nan Li
Keyword(s):  
Recurrence Relation ◽  
Linear Transformation ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Sufficient Condition ◽  
Linear Transformations ◽  
Whitney Numbers ◽  
Three Term Recurrence Relation

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

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.

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