In the joint work of T.Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function $β(s)$ at even positive integers. The construction gives some bonuses for Catalan's constant $G=β(2)$, such as a second-order Apery-like recursion and a permutation group in the sense of G.Rhin and C.Viola. Here we prove expected integrality properties of solutions to the above recursion as well as suggest a simpler (also second-order and Apery-like) one for $G$. We "enlarge" the permutation group of our previous (joint with T.Rivoal) work by showing that the total 120-permutation group for $ζ(2)$ (known thanks to G.Rhin and C.Viola) can be applied in arithmetic study of Catalan's constant. These considerations have computational meanings and do not allow us to prove the (presumed) irrationality of $G$. Finally, we suggest a conjecture yielding the irrationality property of numbers (e.g., of Catalan's constant) from existence of suitable second-order difference equations (recursions).
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.
We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.
Based on a $q$-congruence of the author and Petrov, we set up a $q$-analogue of Sun--Tauraso's congruence for sums of Catalan numbers, which extends a $q$-congruence due to Tauraso.
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…
In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving zeta(3), pi, and Catalan's constant. They indicate that they were not able to deduce these products directly from the values of a function introduced in 1993 by Borwein and Dykshoorn. We provide here such a proof for two of these formulas. We also give a direct proof for the other two formulas, by using a generalization of the Borwein-Dykshoorn function due to Adamchik. Finally we give an expression of the Borwein-Dykshoorn function in terms of the "parameterized-Euler-constant function" introduced b…
Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let $S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$. We also determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$ for $m=7,16,25,32,64,160,800,1600, 156832$.
The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences: $$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\ \mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\ \mbox{modulo}\ p$$ for any prime $p>3$, the second one was conjectured by Z.-W. Sun in 2012. These two congruences are very important to prove the following conjectures of Z.W.Sun: For any old prime $p$, we have $$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\bi…
In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n, k}=C_{2n,n-k}$ and $A_{n, k}=C_{2n+1,n+1-k}$. In fact, some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n-1}=C_{2n+1,n}=C_n$. We present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that, we check sums (and alternating sums) of squares and cubes of $C_{m,k}$ and, consequently, for $ B_{n, k}$ an…
Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study congruences involving higher-order Catalan numbers $C_k^{(h)}=\binom[(h+1)k,k]/(hk+1)$ and $\bar C_k^{(h)}=\binomal[(h+1)k,k]*h/(k+1)$. Our tools include linear recurrences and the theory of cubic residues. Here are some typical results in the paper. (i) If $p^a=1 (mod 6)$ then $$\sum_{k=1}^{p^a-1}\binom[3k,k]/6^k=2^{(p^a-1)/3}-1 (mod p).$$ Also, $$\sum_{k=0}^{p^a-1}…
In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. A…
The author gives the full list of his conjectures on series for powers of $π$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for $π^{-1}$, four series for $π^2$, two series for $π^{-2}$, four series for $π^4$, two series for $π^5$, three series for $π^6$, seven series for $ζ(3)$, one series for $πζ(3)$, two series for $π^2ζ(3)$, one series for $ζ(3)^2$, three series involving both $ζ(3)^2$ and $π^6$, one series for $ζ(5)$, three series involving both $ζ…
Let $<¶> \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $¶$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < ¶>: n \leq x} \frac{μ(n)}{n}$ are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to $\prod_{p \in ¶} (1 - \frac{1}{p})$ (the case when $¶$ is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of non-trivial zeroes and poles of the associated zeta…
