Related papers: On a congruence involving $q$-Catalan numbers
In this paper, we establish a q-analog of partial fraction decomposition formula. By using formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit…
In this paper we present many congruences for several Ap\'ery-like sequences.
In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.
Catalan-Daehee numbers and polynomials, generating functions of which can be expressed as p-adic Volkenborn integrals on Zp, were studied previously. The aim of this paper is to introduce q-analogues of the catalan-Daehee numbers and…
We give a q-analogue of Gauss' divisibility theorem
In terms of the operator method, we prove two conjectural series for $\pi$ of Sun involving harmonic numbers of order two. Furthermore, we also give $q$-analogues of six $\pi$-formulas including the two ones just mentioned.
Inspired by the recent work of El Bachraoui, we present some new $q$-supercongruences on triple and quadruple sums of basic hypergeometric series. In particular, we give a $q$-supercongruence modulo the fifth power of a cyclotomic…
In this paper, several weighted summation formulas of $q$-hyperharmonic numbers are derived. As special cases, several formulas of hyperharmonic numbers of type $\sum_{\ell=1}^{n} {\ell}^{p} H_{\ell}^{(r)}$ and $\sum_{\ell=0}^{n} {\ell}^{p}…
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…
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms. These objects are enumerated by the well-known Catalan numbers, and have been introduced with the aim of giving a common language to most of the structures…
Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of…
In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the…
We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple…
This text evolves from the lecture notes for my course on Catalan's conjecture in winter term 2025/26. The ultimate goal is to give full details of Mih\u{a}ilescu's proof. Current chapters: 1. Euler's theorem: $x^2-y^3=1$; 2. V. Lebesgue's…
In this paper, we derive formulas for the translated Whitney-Lah numbers and show that they are generalizations of already-existing identities of the classical Lah numbers. q-analogues of the said formulas are also obtained for the case of…
We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad…
We prove several Stern's type congruences for generalized bernoulli numbers.
In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation…