Related papers: "Divergent" Ramanujan-type supercongruences
On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q…
We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are…
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
We establish some supercongruences for the truncated ${}_2F_1$ and ${}_3F_2$ hypergeometric series involving the $p$-adic Gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated ${}_3F_2$…
Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including…
In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors.…
This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalsch\"utz' summation formula in the…
We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.
Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. In this paper we compute the high dimensional Hodge-Laplace spectrum of Ramanujan triangle complexes, and show that it implies a…
The hamiltonian describing a single fermion in a Penning trap is shown to be supersymmetric in certain cases. The supersymmetries of interest occur when the ratio of the cyclotron frequency to the axial frequency is 3/2 and the particle has…
We use invariance theory to compute the divergence term $a_{m+2,m}^{d+\delta}$ in the super trace for the twisted de Rham complex for a closed Riemannian manifold.
In the article it was shown the convergence of special integral of two dimensional Terry's problem. Main tools of the article are an investigation of real algebraic varieties and estimations of areas of algebraic surfaces.
We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the…
We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.
In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in…
For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we…
Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series…
We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a…
We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…