Results 11 to 20 of about 623 (41)

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

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

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

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  

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

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  

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

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

Proofs of some binomial identities using the method of last squares [PDF]

, 2010
We give combinatorial proofs for some identities involving binomial sums that have no closed form.Comment: 8 pages, 16 ...
Shattuck, Mark, Waldhauser, Tamás
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On Zudilin's q-question about Schmidt's problem

, 2012
We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31].
Guo, Victor J. W., Zeng, Jiang
core   +1 more source