Partitions are a major part of the ramanujan story as shown in the new film about his life but what are they. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the master theorem. Page from ramanujan s notebook stating his master theorem. In this paper we prove an analogue of ramanujan s master theorem for the hypergeometric fourier transform on root systems. We study a number of possible extensions of the ramanujan master theorem, which is formulated here by using methods of umbral nature. Ramanujan s notebooks ramanujan s nested radicals mock theta function and ramanujan theta function landau ramanujan constant ramanujan conjecture ramanujan prime ramanujan soldner constant ramanujan s sum roger ramanujan identities master theorem some properties of bernoullis numbers. For the interpolation formula, which is used in proving the master theorem, hardy proved it using the.
Ramanujan s master theorem and twoformulas for zeroorder hankel transform a. The formula has been used in statistical physics and is als. Ramanujan s master theorem is related to analytic functions and mellin transforms. As noted in the comments, berndt discusses ramanujans master theorem and some of its applications in ramanujans notebooks, part i 1985page 297. Some applications of celebrated master theorem of ramanujan m. Ramanujans master theorem for hermitian symmetric spaces. Some applications of celebrated master theorem of ramanujan. Ramanujan master theorem is a technique developed by the indian mathematician s. And, analytic function is a function that is locally given by a convergent power series. Ramanujans master theorem for riemannian symmetric spaces. The critical issue is the extension of the function. Ramanujans formula for the riemann zeta function extended. Ramanujan introduced a technique, known as ramanu jans master theorem, which provides an explicit expression for the. In mathematics, ramanujan s master theorem named after mathematician srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of a function.
Finally, a multidimensional extension of ramanujans master theorem is discussed. Abstract in this short note we use the umbral formalism to derive the ramanujan master theorem and discuss its extension to more general cases. Ramanujan introduced a technique, known as ramanujans master theorem, which provides an explicit expression for the mellin transform of a function in terms of the analytic continuation of. In illustration, i mention the exposition by moll and his colleagues 1 that illustrates various neat applications of ramanujan s master theorem, which extrapolates the taylor coe. Berndt, ramanujan s notebooks, part i, springerverlag, new york, 1985. Ramanujan s master theorem states that, under suitable conditions, the mellin transform of a power series provides an interpolation formula for the coef. Also it welcomes open problems in the line of the aim of this udt for possible publication in this section. In the analysis of algorithms, the master theorem for divideandconquer recurrences provides an asymptotic analysis using big o notation for recurrence relations of types that occur in the analysis of many divide and conquer algorithms. Practice problems and solutions master theorem the master theorem applies to recurrences of the following form. Ramanujan introduced a technique, known as ramanujan s master theorem, which provides an analytic expression for the mellin transform of a function.
Ramanujans master theorem and two formulas for zeroorder. Tomorrows episode is about the selftaught mathematical genius ramanujan, but its also about the beauty of mathematics, the language of the universe. In mathematics, ramanujans master theorem is a technique that provides an analytic. Finally, a multidimensional extension of ramanujan s master theorem is discussed. Ramanujans master theorem and duality of symmetric spaces.
Using ramanujans master theorem, two formulas are derived which define the hankel transforms of order zero with even func tions by inverse. In this paper we generalize ramanujan s master theorem to the context of a siegel domain of type ii, which is the hermitian symmetric space of a real noncompact semisimple lie group of finite center. Ramanujan master theoremcontrollare ed inserire frasi. This phrase actually means that ramanujan was such a natural genius that his contributions to mathematics could reach infinity if he would have lived forever. For the love of physics walter lewin may 16, 2011 duration. What are the real world applications of ramanujans findings. The representations of 1729 as the sum of two cubes appear in the bottom right corner. Shaun cooper massey university, auckland askey 80 conference. This gives us the powerful interpretation of j5 equivalently f as coordinate on the genus 0. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Hardys 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.
Ramanujan graphs constructed in the literature, which we shall refer to below, have been all simple graphs. Functional analysis proof of ramanu jans master theorem. I have full proofs of that in my phd thesis but there are written in greek see researchgate. Ramanujan s formula needs to be slightly corrected, but what is remarkable is that such a formula exists. Keywords ramanujans master theorem infinite series closed form pi formula harmonic number. Ramanujans master theorem, the ramanujan journal 10. Remarks on green function of spacefractional biharmonic. Ramanujan s theories of elliptic functions to alternative bases, and beyond. Boston university, 1988 a thesis submitted in partial fulfillment of the requirements for the degree of master of arts in mathematics the graduate school the university of maine may, 2005. Ramanujan introduced a technique, known as ramanujan s master theorem, which provides an explicit expression for the mellin transform of a function in terms of the analytic continuation of its taylor coefficients. Ramanujan to evaluate a class of definite integrals. Pdf ramanujans master theorem victor moll academia.
This technique is used here to calculate the values of. We discuss the implications of the procedure for the theory of special functions, like the derivation of formulae concerning the integrals of products of families of bessel functions and the successive derivatives of bessel type functions. In mathematics, ramanujan s master theorem named after srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of an analytic function. That is, the spherical transform of a spherical series can be expressed in terms of the coefficients of this series. To make this step todays students need more help than their predecessors did, and must be coached and encouraged more. Functional analysis proof of ramanujans master theorem. Ramanujan introduced a technique, known as ramanujans master theorem, which provides an explicit expression for the mellin transform of a function in terms of the analytic continuation of its taylor coefficients. Ramanujan introduced a technique, known as ramanujans master theorem, which provides an explicit expression for the mellin transform of a. The equation expressing the near counter examples to fermats last theorem appears further up. Ramanujans theories of elliptic functions to alternative. Section 3 briefly outlines hardys proof of ramanujans master theorem. Research open access extension of hardy s class for. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued.
A mellin transform is an integral transform that can be regarded as the multiplicative version of the twosided laplace transform. The approach was first presented by jon bentley, dorothea haken, and james b. A page from ramanujan s notebook showing his master theorem, courtesy of wikipedia. Quraishi2 and ram pal3 1department of applied sciences and humanities, faculty of engineering and technology, jamia millia islamia a central university, new delhi110025, india.
Ramanujan introduced a technique, known as ramanujans master theorem, which provides an explicit expression for the mellin transform. Saxe in 1980, where it was described as a unifying method for solving such. Based on the duality of compact and noncompact reductive riemannian symmetric spaces inside. Ramanujans master theorem and duality of symmetric. Ramanujan and pi since ramanujan s 1987 centennial, much new mathematics has been stimulated by uncanny formulas in ramanujan s notebooks lost and found. Ramanujan s formula for the riemann zeta function extended to lfunctions by kakherine j. Assume function f x fx f x has an expansion of the form. In this short note we use the umbral formalism to derive the ramanujan master theorem and discuss its extension to more general cases. The main identity of this theorem involves the extrapolation of the sequence of coefficients of the integrand, defined originally as a function on n to c. Ramanujans master theorem states that, under suitable conditions, the mellin transform of a power series provides an interpolation formula for. Strauch editorial board welcome papers containing some progress in problems listed below. Extension of hardys class for ramanujans interpolation. An analogue of ramanujans master theorem semantic scholar.
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