Ramanujan's Formula Research Articles

Applications of an identity of Andrews

In this paper, we give a bilateral form of an identity of Andrews, which is a generalization of the 1ψ1 summation formula of Ramanujan. Using Andrews’ identity, we deduce some new identities involving mock theta functions of second order and finally, we deduce some q-gamma, q-beta and eta function identities.

97.24 A constant in a formula of Ramanujan

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BILATERAL ZETA FUNCTIONS AND THEIR APPLICATIONS

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversion formula of the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.

Backwards Induction and a Formula of Ramanujan

SummaryProof by induction is a very versatile technique. A short proof of an elegant formula of Ramanujan involving continued roots provides an opportunity to look at two examples in which induction appears to work backwards.

On a remarkable formula of Ramanujan

A simple proof of Ramanujan’s formula for the Fourier transform of |Γ (a + it)|2 is given where a is fixed and has positive real part and t is real. The result is extended to other values of a by solving an inhomogeneous ODE, and we use it to calculate the jump across the imaginary axis.

On a sequence of algebraic formulae of Ramanujan

Among the many astonishing formulae stated by Ramanujan we find in his Notebooks the following sequence (see [1], p 96, Entry 43):which hold for any triple of complex numbers (a, b, c) such that a + b + c = 0.Ramanujan concluded this list by ‘And so on’, which suggests that he had some kind of method or algorithm allowing him to extend this list to any even power of the quadratic form p(a, b, c) = ab + bc + ca, when a + b + c = 0. To explain this, Bruce Brendt details a theorem by S. Bhargava [2], which can be used to produce identities of the kind above, and infers that Ramanujan is likely to have used the same proof to establish his identities (see [1], pp. 97-100)

Character analogues of Ramanujan-type integrals involving the Riemann Ξ-function

A new class of integrals involving the product of4-functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type F.z;;/ D F. z;; N / D F. z;; N / D F.z;;/; where D 1. New character analogues of the Ramanujan‐Guinand formula, the Koshliakov’s formula, and a transformation formula of Ramanujan, as well as its recent generalization, are shown as particular examples. Finally, character analogues of a conjecture of Ramanujan, and Hardy and Littlewood involving infinite series of Mobius functions are derived.

Extension of Hardy's class for Ramanujan's interpolation formula and master theorem with applications

Hardy's class of functions for the Ramanujan interpolation formula and Ramanujan-Hardy master theorem is extended to a wider domain of applicability and the extension is illustrated by examples.MSC: 11M06, 11M26, 11M36, 11M99, 42 A32, 11M06, 42A32

An improvement of the Ramanujan formula for approximation of the Euler gamma function

The aim of this paper is to improve a double inequality due to Ramanujan for approximation of the Euler gamma function.

Ramanujan’s modular equations and Weber–Ramanujan class invariants G n and g n

In this paper, we use Ramanujan's modular equations and transformation formulas to find several new values of Weber-Ramanujan class invariants gn .W e also give new approach to some known values of class invariants Gn.

Ramanujan’s enigmatic formula for the harmonic series

We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.

A general class of Voronoi's and Koshliakov–Ramanujan's summation formulas involving d k (n)

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained.

ANALOGUES OF A TRANSFORMATION FORMULA OF RAMANUJAN

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma function and Riemann zeta function present in his Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular-type relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A. P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.

Improved asymptotic formulas for the gamma function

The aim of this paper is to improve the Ramanujan formula for approximation the gamma function. A fast asymptotic series is constructed.

QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

Wallis-Ramanujan-Schur-Feynman

One of the earliest examples of analytic representations for π is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula In trying to understand the behavior of this integral when the integrand is replaced by the inverse of a product of distinct quadratic factors, the authors encounter relations to some formulas of Ramanujan, expressions involving Schur functions, and Matsubara sums that have appeared in the context of Feynman diagrams.

Ramanujan formula for the generalized Stirling approximation

The aim of this paper is to improve Ramanujan’s formula for approximation of the factorial function, starting from Burnside’s formula in contradistinction with the classical formula that starts from Stirling’s formula.

Transformation formulas associated with integrals involving the Riemann Ξ-function

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.

Series transformations and integrals involving the Riemann Ξ-function

The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of Riemann's Ξ-function. We give proofs of all of these transformation formulas using the theory of Mellin transforms and the residue theorem. Our study includes new extensions of the formulas of Koshliakov and Ferrar through their connection with integrals involving the Riemann Ξ-function.

Some formulas of Ramanujan involving Bessel functions

Some formulas of Ramanujan involving Bessel functions