Related papers: Some new results about $q$-trinomial coefficients
In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.
In this paper, we introduce three new classes of binomial sums involving Fibonacci (Lucas) numbers and weighted binomial coefficients.
We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term…
Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized…
The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of…
In this paper, we investigate some properties of q-Bernoulli polynomi- als arising from q-umbral calculus. Finally, we derive some interesting identities of q-Bernoulli polynomials from our investigation.
$q$-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, $q$-analogs of various probability distributions have been…
The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the…
In this paper, we derive certain congruences for the number of $3$-core cubic bipartitions using elementary $q$-series manipulations and dissection formulas.
Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan's $q$-hypergeometric series by relating it to real quadratic field $\Q(\sqrt{6})$ and using the arithmetic of $\Q(\sqrt{6})$, hence solved a…
The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an…
The derivative polynomials introduced by Knuth and Buckholtz in their calculations of the tangent and secant numbers are extended to a multivariable $q$--environment. The $n$-th $q$-derivatives of the classical $q$-tangent and $q$-secant…
Using the recent result of Bourgain, Demeter and Guth on Vinogradov's mean value, a number of new results about small fractional parts of polynomials and fractional parts of additive forms are obtained. These improve work of Baker, Cook,…
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an…
Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…
In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results. We also provide the recurrence relations and the…
In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…