Related papers: Analytic expressions for some Mellin transforms wi…
The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as…
Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several…
This paper aims to show that by making use of Ramanujan's Master Theorem and the properties of the lower incomplete gamma function, it is possible to construct a finite Mellin transform for the function $f(x)$ that has infinite series…
Ramanujan's Master Theorem is a decades-old theorem in the theory of Mellin transforms which has wide applications in both mathematics and high energy physics. The unconventional method of Ramanujan in his proof of the theorem left…
S. Ramanujan introduced a technique in 1913 for providing analytic expressions for certain Mellin-type integrals which is now known as Ramanujan's Master Theorem. This technique was communicated through his "Quarterly Reports" and has a…
Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of Riemannian symmetric spaces of compact…
Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…
A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for…
We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…
We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the…
Using Ramanujan's Master Theorem, two formulas are derived which define the Hankel transforms of order zero with even functions by inverse Mellin transforms, provided these functions and their derivatives obey special conditions. Their…
In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class…
This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several…
We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen…
The main objective of this paper is to introduce the modified q-Genocchi polynomials and to define their generating function. In the paper, we show new relations, which are explicit formula, derivative formula, multiplication formula, and…
In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in…
The classical transformation of Jacobi's theta function admits a simple proof by producing an integral representation that yields this invariance apparent. This idea seems to have first appeared in the work of S. Ramanujan. Several examples…
We use the Ramanujan's master theorem to evaluate the integral $$\int_{0}^{\infty}\frac{x^{l-1}}{(1+x)^{m+1}}\log^{n}(1+x)\, dx$$ in terms of the digamma function, the gamma function, and the Hurwitz zeta function.
Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.
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…