Pub Date : 2023-01-05DOI: 10.12988/imf.2023.912382
J. Cereceda
In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers $S_k(n) = 1^k + 2^k + cdots + n^k$ involving simultaneously the Stirling numbers of the first and second kind. In this note, we first recall and then slightly refine Lang's formula for $S_k(n)$. As it turns out, the refined Lang's formula constitutes a special case of a well-known relationship between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions. In addition, we provide several applications of this general relationship.
{"title":"A refinement of Lang's formula for the sums of powers of integers","authors":"J. Cereceda","doi":"10.12988/imf.2023.912382","DOIUrl":"https://doi.org/10.12988/imf.2023.912382","url":null,"abstract":"In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers $S_k(n) = 1^k + 2^k + cdots + n^k$ involving simultaneously the Stirling numbers of the first and second kind. In this note, we first recall and then slightly refine Lang's formula for $S_k(n)$. As it turns out, the refined Lang's formula constitutes a special case of a well-known relationship between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions. In addition, we provide several applications of this general relationship.","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121244754","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}
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $pi$.
{"title":"A method for evaluating definite integrals in terms of special functions with examples","authors":"Robert Reynolds, Allan Stauffer","doi":"10.12988/imf.2020.91272","DOIUrl":"https://doi.org/10.12988/imf.2020.91272","url":null,"abstract":"We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $pi$.","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130219555","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}
A strong version of Andrica's conjecture can be formulated as follows: Except for $p_nin{3,7,13,23,31,113}$, that is $nin{2,4,6,9,11,30}$, one has$sqrt{p_{n+1}}-sqrt{p_n} < frac{1}{2}.$ While a proof is far out of reach I shall show that this strong version of Andrica's conjecture is unconditionally and explicitly verified for all primes below the location of the 81$^{st}$ maximal prime gap, certainly for all primes $p <2^{64}approx 1.844times 10^{19}$. Furthermore this strong Andrica conjecture is slightly stronger than Oppermann's conjecture --- which in turn is slightly stronger than both the strong and standard Legendre conjectures, and the strong and standard Brocard conjectures. Thus the Oppermann conjecture, and strong and standard Legendre conjectures, are all unconditionally and explicitly verified for all primes $p <2^{64}approx1.844times 10^{19}$. Similarly, the strong and standard Brocard conjectures are unconditionally and explicitly verified for all primes $p <2^{32} approx 4.294 times 10^9$.
{"title":"Strong version of Andrica's conjecture","authors":"M. Visser","doi":"10.12988/imf.2019.9729","DOIUrl":"https://doi.org/10.12988/imf.2019.9729","url":null,"abstract":"A strong version of Andrica's conjecture can be formulated as follows: Except for $p_nin{3,7,13,23,31,113}$, that is $nin{2,4,6,9,11,30}$, one has$sqrt{p_{n+1}}-sqrt{p_n} < frac{1}{2}.$ While a proof is far out of reach I shall show that this strong version of Andrica's conjecture is unconditionally and explicitly verified for all primes below the location of the 81$^{st}$ maximal prime gap, certainly for all primes $p <2^{64}approx 1.844times 10^{19}$. Furthermore this strong Andrica conjecture is slightly stronger than Oppermann's conjecture --- which in turn is slightly stronger than both the strong and standard Legendre conjectures, and the strong and standard Brocard conjectures. Thus the Oppermann conjecture, and strong and standard Legendre conjectures, are all unconditionally and explicitly verified for all primes $p <2^{64}approx1.844times 10^{19}$. Similarly, the strong and standard Brocard conjectures are unconditionally and explicitly verified for all primes $p <2^{32} approx 4.294 times 10^9$.","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124783752","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}
Pub Date : 2015-11-23DOI: 10.12988/IMF.2021.912181
E. González, D. Weinberg
For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.
{"title":"Root configurations of real univariate cubics and quartics","authors":"E. González, D. Weinberg","doi":"10.12988/IMF.2021.912181","DOIUrl":"https://doi.org/10.12988/IMF.2021.912181","url":null,"abstract":"For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122461328","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}
L. C. R. P. Watagoda, H. S. R. A. Don, Jose Almer T. Sanqui
We propose a new approximation to the skew normal distribution, a cosine approximation (CASN). This distribution is in a closed form and easy to use. CASN is especially useful in statistical inference as it approximates the tail probabilities with very small absolute errors. Graphical and numerical comparisons are conducted to compare the probability density functions of skew normal and the CASN . Mathematics Subject Classification: 62E17
{"title":"A cosine approximation to the skew normal distribution","authors":"L. C. R. P. Watagoda, H. S. R. A. Don, Jose Almer T. Sanqui","doi":"10.12988/imf.2019.9939","DOIUrl":"https://doi.org/10.12988/imf.2019.9939","url":null,"abstract":"We propose a new approximation to the skew normal distribution, a cosine approximation (CASN). This distribution is in a closed form and easy to use. CASN is especially useful in statistical inference as it approximates the tail probabilities with very small absolute errors. Graphical and numerical comparisons are conducted to compare the probability density functions of skew normal and the CASN . Mathematics Subject Classification: 62E17","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115240143","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 split zero point problem of the system of quasi variational inclusion in Hilbert spaces","authors":"Yuliang Shan, Jian-Qiang Zhang","doi":"10.12988/imf.2019.929","DOIUrl":"https://doi.org/10.12988/imf.2019.929","url":null,"abstract":"","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125179966","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}
Pub Date : 1900-01-01DOI: 10.12988/IMF.2021.912244
I. Nuñez
In this study, we present the function H(x)p based on Pk(x, a) introduced by Lehmer. H(x)p denotes the number of numbers that are not divisible by prime numbers < p but are divisible by p. Herein, we show that H(x)p can be obtained only using x 3 . We also present our own prime counting function based on H(x)p, that is, x 3 . Mathematics Subject Classification: 11A41, 11N05
{"title":"Prime counting function in base of x/3","authors":"I. Nuñez","doi":"10.12988/IMF.2021.912244","DOIUrl":"https://doi.org/10.12988/IMF.2021.912244","url":null,"abstract":"In this study, we present the function H(x)p based on Pk(x, a) introduced by Lehmer. H(x)p denotes the number of numbers that are not divisible by prime numbers < p but are divisible by p. Herein, we show that H(x)p can be obtained only using x 3 . We also present our own prime counting function based on H(x)p, that is, x 3 . Mathematics Subject Classification: 11A41, 11N05","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122672380","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}
Pub Date : 1900-01-01DOI: 10.12988/imf.2022.912317
R. Kantrowitz
The purpose of this article is to offer a short, guided tour through the introduction and development of a class of sequence spaces that represent a discretization of spaces of functions of generalized bounded variation. The paths that we follow are motivated by, and parallel, some of those forged over the last 140 years in expanding Jordan’s original concept of functions of bounded variation on a compact interval. In addition, we devote attention to the issue of stability of the sequence spaces under coordinatewise multiplication and also endow them with a canonical norm.
{"title":"Sequences of generalized bounded variation","authors":"R. Kantrowitz","doi":"10.12988/imf.2022.912317","DOIUrl":"https://doi.org/10.12988/imf.2022.912317","url":null,"abstract":"The purpose of this article is to offer a short, guided tour through the introduction and development of a class of sequence spaces that represent a discretization of spaces of functions of generalized bounded variation. The paths that we follow are motivated by, and parallel, some of those forged over the last 140 years in expanding Jordan’s original concept of functions of bounded variation on a compact interval. In addition, we devote attention to the issue of stability of the sequence spaces under coordinatewise multiplication and also endow them with a canonical norm.","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117296379","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}
In this paper, firstly, we obtained the differential equation of the hyper-asymptotic curves in Wn with respect to a rectilinear congruence. With the help of it, we defined the hyper-asymptotic curvature vector field and the hyper-asymptotic curve in Wn. Secondly, we described an asymptotic line of order p in Wn in Wn+1 and a geodesic of order p in Wn+1. We gave necessary and sufficient condition to be an asymptotic line of a curve in Wn. And then we expressed the relation between geodesics in Wn and in Wn+1. Thirdly, we stated the relations among a hyper-asymptotic curve, an asymptotic line of second order and a geodesic of second order in Wn. Finally, we expressed the condition to be a geodesic of second order of a hyper-asymptotic curve. Mathematics Subject Classification: 53B25, 53A25
{"title":"Hyper-asymptotic curves of a Weyl hypersurface","authors":"N. Kofoğlu","doi":"10.12988/imf.2019.9937","DOIUrl":"https://doi.org/10.12988/imf.2019.9937","url":null,"abstract":"In this paper, firstly, we obtained the differential equation of the hyper-asymptotic curves in Wn with respect to a rectilinear congruence. With the help of it, we defined the hyper-asymptotic curvature vector field and the hyper-asymptotic curve in Wn. Secondly, we described an asymptotic line of order p in Wn in Wn+1 and a geodesic of order p in Wn+1. We gave necessary and sufficient condition to be an asymptotic line of a curve in Wn. And then we expressed the relation between geodesics in Wn and in Wn+1. Thirdly, we stated the relations among a hyper-asymptotic curve, an asymptotic line of second order and a geodesic of second order in Wn. Finally, we expressed the condition to be a geodesic of second order of a hyper-asymptotic curve. Mathematics Subject Classification: 53B25, 53A25","PeriodicalId":107214,"journal":{"name":"International Mathematical Forum","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114772529","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}
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