Related papers: $N$-tuple sum analogues for Ramanujan-type congrue…

We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple…

Motivated by observations of Guillera we generalise the so-called Ramanujan-type supercongruences to a further level in which the sequences of Fibonacci, Lucas, Ap\'ery numbers and their friends all receive a natural appearance.

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$

We present the exact effective superpotentials in $4d$, $N=1$ supersymmetric $SU(2)$ gauge theories with $N_3$ triplets and $N_2$ doublets of matter superfields. For the theories with a single triplet matter superfield we present the exact…

A new sums-of-tails identity involving two parameters $b$ and $d$ is obtained and is used to derive more results of similar type. One of Ramanujan's sums-of-tails identities from the Lost Notebook is shown to be a special case of our…

Example 7, after Entry 43, in Chapter XII of the first Notebook of Srinivasa Ramanujan is proved and, more generally, a summation theorem for $_3F_2(a,a,x;1+a,1+a+N;1)$, where $N$ is a non-negative integer, is derived.

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…

We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for…

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 $p\equiv 3\pmod{4}$,…

An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^2x\equiv a\pmod{r}$. In this paper we introduce an analogue of Ramanujan's sum with respect to regular integers (mod $r$) and show that this analogue…

A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are…

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes.…

In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series

In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs. They are obtained as quotients of the buildings of type $\tilde{A}_{d-1}$ associated with…

A supercongruence is a congruence between rational numbers modulo a power of a prime. In this paper, we give a technique for finding and algorithmically proving supercongruences by expressing terms as infinite series involving certain…

We investigate Newton series for truncated multiple $L$-values and thereby obtain a class of relations for multiple $L$-values. In addition, we give a formulation and a proof of extended derivation relations for multiple $L$-values.

This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in $\mathbb{Z}/q\mathbb{Z}$,…