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The numbers which are sum of first n natural numbers are called Triangular numbers and numbers which are product of two consecutive positive integers are called Pronic numbers. The concept of Ramanujan summation has been dealt by Srinivasa Ramanujan for divergent series of real numbers. In this paper, I will determine the Ramanujan summation for positive integral powers of triangular and Pronic numbers and derive a new compact formula for general case.

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

In this paper, we state some q-analogs of the famous Ramanujan Master Theorem. As an application, we compute some values of Jackson’s q-integrals that involve q-special functions.

A century ago, Srinivasa Ramanujan -- the great self-taught Indian genius of mathematics -- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We further discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory., Comment: 32 pages, LaTex, 6 figures, a number of minor improvements

In this study, we apply the Ramanujan summation method to linear Euler sums. This enables us to establish new reflection formulas for these sums that complete results already achieved in our previous works.; Dans cette étude, on applique le procédé de sommation de Ramanujan aux sommes d'Euler linéaires. Ceci nous permet d'établir de nouvelles relations de réciprocité entre sommes d'Euler parachevant les résultats déjà obtenus dans des travaux antérieurs.

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function., 14 pages, 3 figures

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant $$\gamma $$ and the constant $$\ln 2\pi $$ are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

By the Lagrange–Burmann formula, we provide a new explicit formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. Based on the new explicit formula, we obtain two interesting identities for the Bernoulli numbers.

Already in the 18th century, Euler was known to the following account: ∑n = −1/12 (Ramanujan summation). The above infinite sum of natural numbers applies to certain physical issues, including quantum field theory and string theory (e.g. Casimir effect). From a mathematical point of view, it is associated with the so-called Riemann's zeta function. This article derives the above relationship without referring to the Riemann zeta function and the Ramanujan summation method.

The authors present the power series expansions of the function R ( a ) − B ( a ) at a = 0 and at a = 1 / 2 , show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R ( a ) and the beta function B ( a ) ≡ B ( a , 1 − a ) , and obtain asymptotically sharp lower and upper bounds for R ( a ) in terms of B ( a ) and polynomials. In addition, some properties of the Riemann zeta function ζ ( n ) , n ∈ N , and its related sums are derived.

In this letter, a closed-form orthogonal Ramanujan integer basis is proposed and obtained by performing Gram–Schmidt process from the Ramanujan sum and its circular shift. It has a surprisingly simple and sparse form, which is better than the original complete Ramanujan basis. The relationship between the original basis and proposed basis is also clearly illustrated. Therefore, the original basis can be replaced easily.

The purpose of this paper is to give some sequences that converge quickly to Ioachimescu’s constant related to Ramanujan’s formula by the multiple-correction method.

In this paper, we establish some new P-Q type modular equations, by using the modular equations given by Srinivasa Ramanujan.

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers., Comment: 16 pages

Arithmetical functions with absolutely convergent Ramanujan expansions have been recently studied in certain contexts, by the present author, Murty and many others. In this article, we aim to weaken some of their hypotheses and derive asymptotic formulas with an explicit error term for sums of the form [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two arithmetical functions with absolutely convergent Ramanujan expansion.

The aim of this note is to improve an approximation formula of Ramanujan-type discussed by Hirschhorn and Villarino.

Using certain representations for Eisenstein series and Ramanujan's quartic theory of elliptic functions, we establish some new series for 1 / π 2 .

We evaluate in closed form certain classes of infinite integrals containing trigonometric and hyperbolic trigonometric functions in their integrands. Although not apparent, the integrals are related to Jacobian elliptic functions, in particular, to theorems found in Ramanujan’s notebooks. One of our evaluations answers a question first posed by M. E. H. Ismail.

In 2002, L.N. Trefethen of Oxford University challenged the scientific community by ten intriguing mathematical problems to be solved, numerically, to ten digit accuracy each (I was one of the successful contestants; in 2004, jointly with three others of them, I published a book—Bornemann et al.: The SIAM 100-Digit Challenge, SIAM, Philadelphia, 2004—on the manifold ways of solving those problems). In this paper, I collect some new and noteworthy insights and developments that have evolved around those problems in the last decade or so. In the course of my tales, I will touch mathematical topics as diverse as divergent series, Ramanujan summation, low-rank approximations of functions, hybrid numeric-symbolic computation, singular moduli, self-avoiding random walks, the Riemann prime counting function, and winding numbers of planar Brownian motion. As was already the intention of the book, I hope to encourage the reader to take a broad view of mathematics, since one lasting moral of Trefethen’s contest is that overspecialization will provide too narrow a view for one with a serious interest in computation.

We employ Hardy’s regularly convergent double series to refine an argu- ment of Nanjundiah [7]. In particular, we evaluate some alternating series.

In the paper, the authors supply alternative proofs for some summation formulas of rigonometric series.

When studying fluctuations of models in the 1D KPZ class including the ASEP and the $q$-TASEP, a standard approach has been to first write down a formula for $q$-deformed moments and constitute their generating function. This works well for the step initial condition, but there is a difficulty for a random initial condition (including the stationary case): in this case only the first few moments are finite and the rest diverge. In a previous work [16], we presented a method dealing directly with the $q$-deformed Laplace transform of an observable, in which the above difficulty does not appear. There the Ramanujan's summation formula and the Cauchy determinant for the theta functions play an important role. In this note, we give an alternative approach for the $q$-TASEP without using them., 20 pages

Inspired by the recent work by Brychkov and Saad, who establish some infinite summation formulas of Appell's functions F1, F2 and F3. The author, with the help of some famous summation theorems, obtains several infinite summation formulas of double hypergeometric functions, and the infinite summation formulas of Appell and Horn's functions are presented as examples.

In the paper A new parameter for Ramanujan's theta-functions and explicit values, Arab J. Math. Sc., 18 (2012), 105-119, Saikia studied the parameter involving Ramanujan's theta-functions and for any positive real numbers k and n and applied it to find explicit values of . As more application to the parameter , in this paper we prove a new general theorem for explicit evaluation of Ramanujan--Gordon continued fraction K(q) in terms of the parameter and give examples. We also find some new explicit values of the parameter and offer reciprocity theorems for the continued fraction K(q).

We briefly review Ramanujan’s theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for 1/π similar to ones found by Ramanujan.

In this paper, we give an alternative proof of a Ramanujan's modular equation of degree 7 by employing certain theta function identities.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

Ramanujan proved that the inequality holds for all sufficiently large values of x. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that it holds if x ≥ exp (9658). Furthermore, we solve the inequality completely on the Riemann hypothesis and show that x = 38 358 837 682 is the largest integer counterexample.

Starting from the classical summation formulas and basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of new summation formulas of the Poisson, Muntz, Mobius and Voronoi type. The corresponding analogues of the Muntz operators are investigated. Interesting and curious particular cases of summation formulas involving arithmetic functions are exhibited. Necessary and sufficient conditions for the validity of the Riemann hypothesis are derived.