{"title":"Location of zeros of the polar derivative of a polynomial","authors":"B. Zargar, M. Gulzar, S. A. Malik","doi":"10.7153/jca-2022-19-01","DOIUrl":"https://doi.org/10.7153/jca-2022-19-01","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on a family of log-integrals","authors":"Khristo N. Boyadzhiev, R. Frontczak","doi":"10.7153/jca-2022-20-10","DOIUrl":"https://doi.org/10.7153/jca-2022-20-10","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An elementary proof of Ramanujan's identity for odd zeta values","authors":"Sarth Chavan","doi":"10.7153/jca-2022-19-11","DOIUrl":"https://doi.org/10.7153/jca-2022-19-11","url":null,"abstract":"The Riemann zeta function ζ(s) is one of the most important special functions of Mathematics. While the critical strip 0 < R (s) < 1 is undoubtedly the most important region in the complex plane on account of the unsolved problem regarding location of non-trivial zeros of ζ(s), namely, the Riemann Hypothesis, the right-half plane R (s) > 1 also has its own share of interesting unsolved problems to contribute to. It is quite well known that many number theoretic properties of odd zeta values are nowadays still unsolved mysteries, such as the rationality, transcendence and existence of closed-forms. Only in 1978 did Apéry [2] famously proved that ζ(3) is irrational. This was later reproved in a variety of ways by several authors, in particular Beukers [10] who devised a simple approach involving certain integrals over [0, 1]. In the early 2000s, an important work of Rivoal [21], and Ball and Rivoal [4] determined that infinitely many values of ζ at odd integers are irrational, and the work of Zudilin [27] proved that at least one among ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. A very recent result due to Rivoal and Zudilin [22] states that at least two of the numbers ζ(5), ζ(7), . . . , ζ(69) are irrational. Moreover, for any pair of positive integers a and b, Haynes and Zudilin [17, Theorem 1] have shown that either there are infinitely many m ∈ N for which ζ(am+ b) is irrational, or the sequence {qm} ∞ m=1 of common denominators of the rational elements of the set {ζ(a+ b), . . . , ζ(am+ b)} grows super-exponentially, that is, q 1/m m → ∞ as m → ∞. Despite these advances, to this day no value of ζ(2n+ 1) with n > 2 is known to be irrational. A folklore conjecture states that the numbers π, ζ(3), ζ(5), ζ(7), . . . are algebraically independent over the rationals. This conjecture is predicted by the Grothendieck’s period conjecture for mixed Tate motives. But both conjectures are far out of reach and we do not even know the transcendence of a single odd zeta value. One should mention that Brown [11] has in the past few years outlined a simple geometric approach to understand the structures involved in Beukers’s proof of irrationality of ζ(3) and how this may generalize to other odd zeta values. Ramanujan made many beautiful and elegant discoveries in his short life of 32 years. One of the most remarkable formulas suggested by Ramanujan that has attracted the attention of several mathematicians over the years is the following intriguing identity involving the odd values of the Riemann zeta function [6, 1.2]:","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49594737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Necessary and sufficient conditions for the convergence of positive series","authors":"V. Abramov","doi":"10.7153/jca-2022-19-09","DOIUrl":"https://doi.org/10.7153/jca-2022-19-09","url":null,"abstract":"We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran–De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452457] and [L. Bourchtein et al, Int. J. Math. Anal. 6(2012), 1847–1869]. The obtained result enables us to extend the known conditions for recurrence and transience of birth-and-death processes given in [V. M. Abramov, Amer. Math. Monthly 127(2020) 444–448].","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41922378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform norm estimates of Bernstein-type for lacunary-type complex polyomials","authors":"A. Mir, A. Hussain","doi":"10.7153/jca-2021-18-04","DOIUrl":"https://doi.org/10.7153/jca-2021-18-04","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integrating the tails of two Maclaurin series","authors":"Russell A. Gordon","doi":"10.7153/jca-2021-18-06","DOIUrl":"https://doi.org/10.7153/jca-2021-18-06","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On some inequalities concerning generalized (α,β) relative order and generalized (α,β) relative type of entire function with respect to an entire function","authors":"T. Biswas, C. Biswas","doi":"10.7153/jca-2021-18-07","DOIUrl":"https://doi.org/10.7153/jca-2021-18-07","url":null,"abstract":"In this paper, we intend to find out some inequalities relating to generalized (α ,β) relative order, generalized (α ,β) relative type and generalized (α ,β) relative weak type of an entire function f with respect to an entire function g when generalized (γ ,β) relative order, generalized (γ ,β) relative type and generalized (γ ,β) relative weak type of f with respect to another entire function h and generalized (γ ,α) relative order, generalized (γ ,α) relative type and generalized (γ ,α) relative weak type of g with respect to h are given, where α , β and γ are continuous non-negative slowly increasing functions defined on (−∞,+∞) . Mathematics subject classification (2020): 30D20, 30D30, 30D35.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the generalized Hurwitz-Lerch zeta function and generalized Lambert transform","authors":"Viren ra Kumar","doi":"10.7153/jca-2021-17-05","DOIUrl":"https://doi.org/10.7153/jca-2021-17-05","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Alternating Euler sums and BBP-type series","authors":"A. Sofo","doi":"10.7153/jca-2021-18-12","DOIUrl":"https://doi.org/10.7153/jca-2021-18-12","url":null,"abstract":". An investigation into a family of de fi nite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On weighted β-absolute convergence of double Fourier series","authors":"K. N. Darji, R. Vyas","doi":"10.7153/jca-2021-18-01","DOIUrl":"https://doi.org/10.7153/jca-2021-18-01","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}