26 research outputs found
Extensions of Watson's theorem and the Ramanujan-Guinand formula
No full text© 2023 World Scientific Publishing Company.Ramanujan provided several results involving the modified Bessel function Kz(x) in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eisenstein series on SL2(ℤ). Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In this paper, we first obtain a generalization of a theorem of Watson and, as an application of it, give a new proof of the result of Dixit, Kesarwani, and Moll. Watson's theorem is also generalized in a different direction using muλ) which is itself a generalization of Kz(x). Analytic continuation of all these results are also given.11Nsciescopu
Explicit transformations of certain Lambert series
No full textAn exact transformation, which we call the \emph{master identity}, is
obtained for the first time for the series
for and Re.
New modular-type transformations when is a non-zero even integer are
obtained as its special cases. The precise obstruction to modularity is
explicitly seen in these transformations. These include a novel companion to
Ramanujan's famous formula for . The Wigert-Bellman identity
arising from the case of the master identity is derived too. When is
an odd integer, the well-known modular transformations of the Eisenstein series
on , that of the Dedekind eta function
as well as Ramanujan's formula for are derived from the master
identity. The latter identity itself is derived using Guinand's version of the
Vorono\"{\dotlessi} summation formula and an integral evaluation of
N.~S.~Koshliakov involving a generalization of the modified Bessel function
. Koshliakov's integral evaluation is proved for the first time. It
is then generalized using a well-known kernel of Watson to obtain an
interesting two-variable generalization of the modified Bessel function. This
generalization allows us to obtain a new modular-type transformation involving
the sums-of-squares function . Some results on functions
self-reciprocal in the Watson kernel are also obtained.Comment: The earlier title of the paper is modified to the current one. This
paper has now been accepted for publication in 'Research in the Mathematical
Sciences
The generalized modified Bessel function Kz,w(x) at z=1/2 and Humbert functions,
No full textRecently Dixit, Kesarwani, and Moll introduced a generalization Kz,w(x) of the modified Bessel function Kz(x) and showed that it satisfies an elegant theory similar to Kz(x). In this paper, we show that while K12(x) is an elementary function, K12,w(x) can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function ?(z)
Surface plasmon resonance and nonlinear optical behavior of pulsed laser-deposited semitransparent nanostructured copper thin films
No full textApplications of Lipschitz summation formula and a generalization of Raabe's cosine transform
No full textGeneral summation formulas have been proved to be very useful in number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series Ek(z) on SL2(Z), where z?-1/z,z?H. The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with Kesarwani transforming ??n=1?2m(n)e-ny is also derived using the Lipschitz summation formula. In this pursuit, we naturally encounter a new generalization of Raabe's cosine transform. Several of its properties are also demonstrated
An Analytical study between various Indices of Acute Pancreatitis
No full textIntroduction :Acute pancreatitis is a relatively common disease with wide clinical variation and its incidence is increasing. The severity of acute pancreatitis can be predicted based upon various severity grading systems . Some of these can be performed on admission to assist in triage of patients while others can only be obtained during the first 48 to 72 hours or later. The objective of the study was to determine the relation between various indices of Acute pancreatitis. Methods:This observational study involved Prior Consent from the patients & was found to be within ethical standards . A total of 100 patients were selected which were proven cases of Acute Pancreatitis during a period of 2.5 years from year 2016 to 2018. Subjects included both the genders , all age groups including pediatric and geriatric age group and all classes of socio economic strata.Results: Majority of the patients were in the age group of 21-40. There was a clear gender predilection towards males with M:F of 4:1.In all cases ,Pancreatic enzymes showed more than threefold higher than the upper limit of normal. 72% of the cases were related to alcoholism.88% cases were having mild ranson’s score.As per CT Severity index maximum cases were mild in severity. Conclusion:There was a significant male preponderance . Most common cause was alcohol abuse in males and gall stone disease in females. There was a good correlation between Balthazar CT severity index and Ranson‟s score. Magnitude of enzyme elevation had no relation to the severity of the disease. Irrespective of the cause enzyme elevations were similar quantitatively
A Cross sectional Analytical study to find out the magnitude of Microalbuminuria in Patients of type 2 Diabetes Mellitus
No full textIntroduction: Type 2 diabetes mellitus is associated with significant morbidity and mortality mainly due to cardiovascular complications. Microvascular complications, such as diabetic nephropathy and retinopathy are common. Abnormal levels of urinary albumin excretion are seen in 30-40% of diabetics and is a commonest cause of end stage renal disease. This study was aimed to determine the prevalence of microalbuminuria in type-2 diabetic patients and to evaluate the relation between microalbuminuria and age, sex, duration of diabetes, body mass index.Methods: This Cross sectional study involved 100 Subjects of both the genders aged 20 years to 70 years and all classes of socio economic strata attending various local tertiary care hospitals including our Institute . Randomization was done . All patients were subjected to detail history after taking written and informed consent and detail systemic examination. They were subjected to detailed history and physical examination (including vitals, weight, height, and body mass index [BMI]), with special emphasis on the examination of cardiovascular system.
Results: Out of the 100 patients, 61 % of the patients had normal albuminuria and 39% of them had microalbuminuria. Male patients were more in both normoalbuminuria and microalbuminuria. Mean age of detection of diabetes among study population was in the early 40s, but the age when microalbuminuria was detected was a little higher. Body mass index was higher in patients with microalbuminuria. Blood pressure was higher among the patients with microalbuminuria compared to normal albuminuria patients. In this study, the biochemical parameters were on the higher among microalbuminuria patients . Neuropathy was commonest complication in both group of patients followed by NPDR in patients with microalbuminuria Conclusion: Prevalence of microalbuminuria was seen in patients with type 2 diabetes. Hypertension, raised HbA1C levels, high blood sugar levels and creatinine clearance levels are the major risk factors. Hence early detection of high risk patients and the early initiation of renal and cardiovascular protective agents helps in reducing morbidity and mortality due to type 2 diabetes mellites
Plasmonic interaction between copper nanoparticles and metallic single walled carbon nanotubes
No full text4.5 A 2-phase resonant switched-capacitor converter delivering 4.3W at 0.6W/mm<sup>2</sup> with 85% efficiency
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