Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved...

In this paper, we establish some relations involving q-Euler type sums, q-harmonic numbers and q-polylogarithms. Then, using the relations obtained with the help of q-analog of partial fraction decomposition formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit formulas for several classes of q-harmon...

We give a q-analogue of some binomial coefficient identities of Y. Sun [Electron.

We give some identities involving sums of powers of the partial sum of q-binomial coefficients, which are q-analogues of Hirschhorn’s identities [Discrete Math. 159 (1996), 273–278] and Zhang’s identity [Discrete Math. 196 (1999), 291–298].

This paper presents the evaluation of Euler sums generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr in terms famous harmonic numbers. Moreover, several infinite series, whose consist certain and reciprocal binomial coefficients, are evaluated Riemann zeta values.

The q-analogue of Dixon’s identity involves three q-binomial coefficients as summands. We find many variations of it that have beautiful corollories in terms of Fibonomial sums. Proofs involve either several instances of the q-Dixon formula itself or are “mechanical,” i. e., use the q-Zeilberger algorithm

We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically.