arXiv subjects
Publications and source records attributed to Ji-Cai Liu.
We prove two supercongruences for specific truncated hypergeometric series. These include an uniparametric extension of a supercongruence that was recently established by Long and Ramakrishna. Our proofs involve special instances of various hypergeometric identities including Whipple's transformation and the Karlsson--Minton summation.
In this paper, we calculate the quantum time delays for neutron scattering off the Earth's linear gravitational potential. The quantum time delays are obtained by subtracting the classical returning time (CRT) from the Wigner time, the dwell time and the redefined Larmor time respectively. Different from the conventional definition, our Larmor time is defined by aligning the magnetic field along the neutron propagation direction, and this definition does give reasonable results for motions through a free region and a square barrier. It is worth noting that in the zero magnetic field limit, the Larmor time coincides well with the CRT, which is due to the special shape of linear barrier, and may have some relevance to the weak equivalence principle. It is also found that the classical forbidden region plays an essential role for the dwell time $τ_{_\mathrm{DW}}$ to match with the CRT, and the difference between the dwell and the phase times, \ie, the self-interference time delay, is barrier shape sensitive and clearly shows the peculiarity of the linear barrier. All the time delays are on the order of sub-millisecond and exhibit oscillating behaviors, signaling the self-interference of the scattering neutron, and the oscillations become evident only when the de Broglie wavelength $λ_k=2π/k$ is comparable to the characteristic length $L_c=[2m^2g/\hbar^2]^{-1/3}$. If the time delay measurement is experimentally realizable, it can probe the quantum nature for particle scattering off the gravitational potential in the temporal domain.
We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime $p\ge 5$, \begin{align*} \sum_{k=0}^{p-1}\frac{3k+1}{(-32)^k}{\rm Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3} \pmod{p^4}, \end{align*} which is regarded as a $p$-adic analogue of the following interesting formula for $1/π$ due to Rogers: \begin{align*} \sum_{k=0}^{\infty}\frac{3k+1}{(-32)^k}{\rm Domb}(k)=\frac{2}π. \end{align*} Here ${\rm Domb}(n)$ and $E_n$ are the famous Domb numbers and Euler numbers.
We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence: \begin{align*} \sum_{k=0}^{p-1} \frac{4k+1}{81^k}γ_k \equiv \left(\frac{-3}{p}\right) p\pmod{p^3}, \end{align*} which is corresponding to Chan and Verrill's formula for $1/π$: \begin{align*} \sum_{k=0}^\infty \frac{4k+1}{81^k}γ_k = \frac{3\sqrt{3}}{2π}. \end{align*} Here $γ_n$ are the Almkvist-Zudilin numbers.
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by a spatially uniform dislocation density field in a body comprising a nonlinear elastic material may fail to vanish under no loads. The class of counterexamples constructed in [1] is essentially $2$-dimensional: it works with the subgroup $\mathcal{O}(2) \oplus \langle{\bf Id}\rangle \subset \mathcal{O}(3)$. The objective of this note is to extend Acharya's result in [1] to $\mathcal{O}(3)$, subject to an additional structural assumption and less regularity requirements.
We investigate a variation of $q$-Wolstenholme's theorem, which extends the $q$-analogue of Wolstenholme's theorem due to Shi and Pan [Amer. Math. Monthly 114 (2007), 529--531]. The proof makes use of the Ramanujan sum and higher order logarithmic derivatives of cyclotomic polynomials.
Via symbolic summation method, we establish the following series for $π^2$: \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} = \frac{π^2}{18}, \end{align*} where $H_k=\sum_{j=1}^k 1/j$. We also derive a $p$-adic congruence related to this series. As an application, we prove two congruences involving Franel numbers, one of which was originally conjectured by Sun.
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.
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured by Guo and Schlosser and recently confirmed by Jana and Kalita.
In this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.
We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\ \sum_{k=0}^{(p+1)/2} (4k-1)^5 \frac{(-\frac{1}{2})_k^4}{k!^4} &\equiv 16p\pmod{p^{4}}, \end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\tt Sigma} developed by Schneider. We also put forward two related conjectures.
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.
We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and the other is regarded as $p$-adic analogue of an identity due to Ramanujan.
In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to supercongruences modulo $p^4$ for primes $p\equiv 3\pmod{4}$, which appear to be new.
In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.
In this paper, we prove two conjectural supercongruences on the $(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.