Related papers: On two supercongruences of double binomial sums
In this paper, we prove two conjectural supercongruences on the $(p-1)$th Ap\'ery number, which were recently proposed by Z.-H. Sun.
In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Ap\'ery-like numbers modulo $p^3$, where $p$ is an odd prime.
We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of…
In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…
Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to…
Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…
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 prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv…
We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.
We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…
Recently, Andrews and Bachraoui investigated congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan type congruences and a vanishing identity for the limiting sequence. Very recently, Banerjee,…
Recently, B\'{e}nyi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various…
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…
We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves…
We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a conjectural supercongruence for numbers which are coefficients in one…