Results 21 to 30 of about 686 (63)

A q-rious positivity

, 2010
The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients.
Warnaar, S. Ole, Zudilin, Wadim
core   +1 more source

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

, 2012
For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+
Calkin N. J., Foata D., Franel J., Franel J., Guo V. J.W., Koepf W., MacMachon P. A., Riordan J., Strehl V., Sun Z.-W., Victor J.W. Guo, Zagier D.   +11 more
core   +1 more source

Sums of quadratic half integer harmonic numbers of alternating type

, 2016
Half integer values of quadratic harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed form representations of double integral expressions are developed in terms of special functions.
A. Sofo
semanticscholar   +1 more source

On three-dimensional q-Riordan arrays

Demonstratio Mathematica
In this article, we define three-dimensional q-Riordan arrays and q-Riordan representations for these arrays. Also, we give four cases of infinite multiplication three-dimensional matrices of these arrays.
Fang Gang, Koparal Sibel, Ömür Neşe, Duran Ömer, Khan Waseem Ahmad   +4 more
doaj   +1 more source

Modular forms, hypergeometric functions and congruences

, 2013
Using the theory of Stienstra and Beukers, we prove various elementary congruences for the numbers \sum \binom{2i_1}{i_1}^2\binom{2i_2}{i_2}^2...\binom{2i_k}{i_k}^2, where k,n \in N, and the summation is over the integers i_1, i_2, ...i_k >= 0 such that ...
Kazalicki, M.
core   +2 more sources

Some vanishing sums involving binomial coefficients in the denominator [PDF]

, 2008
Identities involving binomial coeffcients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace [1] using probability theory turn out to be special cases of the Chu ...
Purkait, S. (Soma), Sury, B.
core  

A new family of multivalent functions defined by certain forms of the quantum integral operator

Demonstratio Mathematica
In this work, using the concepts of qq-calculus, we first define the qq-Jung-Kim-Srivastava and qq-Bernardi integral operators for multivalent functions. Then, we use these operators to establish the generalized integral operator ℬq,p−m−λf(z){{\mathcal{ {
Khan Ajmal, Al-shbeil Isra, Shatarah Amani, Alrayes Norah Mousa, Khan Shahid, ul Haq Wasim   +5 more
doaj   +1 more source

Periodic Sequences modulo $m$ [PDF]

, 2015
We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell(m,x)={\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor+b_i}_i}=m{\displaystyle ...
Laugier, Alexandre, Saikia, Manjil
core  

Some congruences involving binomial coefficients

, 2015
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant ...
Cao, Hui-Qin, Sun, Zhi-Wei
core   +1 more source

Ramanujan-type formulae for $1/\pi$: $q$-analogues

, 2018
The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of $\pi$, Archimedes' constant, remain an attractive object of arithmetic study.
Guo, Victor J. W., Zudilin, Wadim
core   +2 more sources