In this article, we define a special function called the Bigamma function. It provides a generalization of Euler's gamma function. Several algebraic properties of this new function are studied. In particular, results linking this new function to the standard Beta function have been provided. We have also established inequalities, which allow to approximate this new function.
{"title":"The BiGamma Function and some of its Related Inequalities","authors":"Raissoulii, Mohamed Chergui","doi":"arxiv-2405.19368","DOIUrl":"https://doi.org/arxiv-2405.19368","url":null,"abstract":"In this article, we define a special function called the Bigamma function. It\u0000provides a generalization of Euler's gamma function. Several algebraic\u0000properties of this new function are studied. In particular, results linking\u0000this new function to the standard Beta function have been provided. We have\u0000also established inequalities, which allow to approximate this new function.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197279","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 the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize corresponding known conclusions.
{"title":"Decreasing and complete monotonicity of two functions defined by three derivatives of a completely monotonic function involving the trigamma function","authors":"Hong-Ping Yin, Ling-Xiong Han, Feng Qi","doi":"arxiv-2405.19361","DOIUrl":"https://doi.org/arxiv-2405.19361","url":null,"abstract":"In the paper, by convolution theorem of the Laplace transforms, a\u0000monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem\u0000for completely monotonic functions, and other analytic techniques, the authors\u0000verify decreasing property of a ratio between three derivatives of a function\u0000involving trigamma function and find necessary and sufficient conditions for a\u0000function defined by three derivatives of a function involving trigamma function\u0000to be completely monotonic. These results confirm previous guesses posed by Qi\u0000and generalize corresponding known conclusions.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189206","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}
Let $ k geq 2 $ be an integer. The $ k- $generalized Fibonacci sequence is a sequence defined by the recurrence relation $ F_{n}^{(k)}=F_{n-1}^{(k)} + cdots + F_{n-k}^{(k)}$ for all $ n geq 2$ with the initial values $ F_{i}^{(k)}=0 $ for $ i=2-k, ldots, 0 $ and $ F_{1}^{(k)}=1.$ In 2020, Banks and Luca, among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem by taking into account $ k-$generalized Fibonacci numbers as concatenations of two terms of the same sequence. We completely solve this problem for all $ k geq 3.
{"title":"On Concatenations of Two $ k $-Generalized Fibonacci Numbers","authors":"Alaa Altassan, Murat Alan","doi":"arxiv-2405.15001","DOIUrl":"https://doi.org/arxiv-2405.15001","url":null,"abstract":"Let $ k geq 2 $ be an integer. The $ k- $generalized Fibonacci sequence is a\u0000sequence defined by the recurrence relation $ F_{n}^{(k)}=F_{n-1}^{(k)} +\u0000cdots + F_{n-k}^{(k)}$ for all $ n geq 2$ with the initial values $\u0000F_{i}^{(k)}=0 $ for $ i=2-k, ldots, 0 $ and $ F_{1}^{(k)}=1.$ In 2020, Banks\u0000and Luca, among other things, determined all Fibonacci numbers which are\u0000concatenations of two Fibonacci numbers. In this paper, we consider the\u0000analogue of this problem by taking into account $ k-$generalized Fibonacci\u0000numbers as concatenations of two terms of the same sequence. We completely\u0000solve this problem for all $ k geq 3.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170201","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}
Tommaso Flaminio, Lluis Godo, Paula Menchón, Ricardo O. Rodriguez
The present paper is devoted to study the effect of connected and disconnected rotations of G"odel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable G"odel algebras endowed with modal operators.
{"title":"Rotations of Gödel algebras with modal operators","authors":"Tommaso Flaminio, Lluis Godo, Paula Menchón, Ricardo O. Rodriguez","doi":"arxiv-2405.19354","DOIUrl":"https://doi.org/arxiv-2405.19354","url":null,"abstract":"The present paper is devoted to study the effect of connected and\u0000disconnected rotations of G\"odel algebras with operators grounded on directly\u0000indecomposable structures. The structures resulting from this construction we\u0000will present are nilpotent minimum (with or without negation fixpoint,\u0000depending on whether the rotation is connected or disconnected) with special\u0000modal operators defined on a directly indecomposable algebra. In this paper we\u0000will present a (quasi-)equational definition of these latter structures. Our\u0000main results show that directly indecomposable nilpotent minimum algebras (with\u0000or without negation fixpoint) with modal operators are fully characterized as\u0000connected and disconnected rotations of directly indecomposable G\"odel\u0000algebras endowed with modal operators.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197275","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}
Abdulhafeez A. Abdulsalam, Ammar K. Mohammed, Hemza Djahel
In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This method allows us to provide new integral representations for the error function. Following this, we use the Fourier transform to derive formulas for the Glasser and Widder potential transforms, leading to several new and interesting corollaries. As part of the applications, we demonstrate the use of one of these integral formulas to provide a new real analytic proof of Euler's reflection formula for the gamma function. Of particular interest is a generalized integral involving the Riemann zeta function, which we also present as an application.
{"title":"New identities for the Laplace, Glasser, and Widder potential transforms and their applications","authors":"Abdulhafeez A. Abdulsalam, Ammar K. Mohammed, Hemza Djahel","doi":"arxiv-2405.14248","DOIUrl":"https://doi.org/arxiv-2405.14248","url":null,"abstract":"In this paper, we begin by applying the Laplace transform to derive closed\u0000forms for several challenging integrals that seem nearly impossible to\u0000evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 =\u0000c^2$, these closed forms become even more intriguing. This method allows us to\u0000provide new integral representations for the error function. Following this, we\u0000use the Fourier transform to derive formulas for the Glasser and Widder\u0000potential transforms, leading to several new and interesting corollaries. As\u0000part of the applications, we demonstrate the use of one of these integral\u0000formulas to provide a new real analytic proof of Euler's reflection formula for\u0000the gamma function. Of particular interest is a generalized integral involving\u0000the Riemann zeta function, which we also present as an application.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146263","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 recent works we have introduced the parameter space $mathcal{Z}_N$ of $A$-variations of the Hardy $Z$-function, $Z(t)$, whose elements are functions of the form begin{equation} label{eq:Z-sections} Z_N(t ; overline{a} ) = cos(theta(t))+ sum_{k=1}^{N} frac{a_k}{sqrt{k+1} } cos ( theta (t) - ln(k+1) t), end{equation} where $overline{a} = (a_1,...,a_N) in mathbb{R}^N$. The ( A )-philosophy advocates that studying the discriminant hypersurface forming within such parameter spaces, often reveals essential insights about the original mathematical object and its zeros. In this paper we apply the $A$-philosophy to our space $mathcal{Z}_N$ by introducing ( Delta_n(overline{a} ) ) the $n$-th Gram discriminant of ( Z(t) ). We show that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law [ (-1)^n Delta_n(overline{1}) > 0, ] for any $n in mathbb{Z}$. We further show that the classical Gram's law ( (-1)^n Z(g_n) >0) can be considered as a first-order approximation of our corrected law. The second-order approximation of $Delta_n (overline{a})$ is then shown to be related to shifts of Gram points along the ( t )-axis. This leads to the discovery of a new, previously unobserved, repulsion phenomena [ left| Z'(g_n) right| > 4 left| Z(g_n) right|, ] for bad Gram points $g_n$ whose consecutive neighbours $g_{n pm 1}$ are good. Our analysis of the (A)-variation space (mathcal{Z}_N) introduces a wealth of new results on the zeros of (Z(t)), casting new light on classical questions such as Gram's law, the Montgomery pair-correlation conjecture, and the RH, and also unveils previously unknown fundamental properties.
{"title":"The $A$-philosophy for the Hardy $Z$-Function","authors":"Yochay Jerby","doi":"arxiv-2406.06548","DOIUrl":"https://doi.org/arxiv-2406.06548","url":null,"abstract":"In recent works we have introduced the parameter space $mathcal{Z}_N$ of\u0000$A$-variations of the Hardy $Z$-function, $Z(t)$, whose elements are functions\u0000of the form begin{equation} label{eq:Z-sections} Z_N(t ; overline{a} ) =\u0000cos(theta(t))+ sum_{k=1}^{N} frac{a_k}{sqrt{k+1} } cos ( theta (t) -\u0000ln(k+1) t), end{equation} where $overline{a} = (a_1,...,a_N) in\u0000mathbb{R}^N$. The ( A )-philosophy advocates that studying the discriminant\u0000hypersurface forming within such parameter spaces, often reveals essential\u0000insights about the original mathematical object and its zeros. In this paper we\u0000apply the $A$-philosophy to our space $mathcal{Z}_N$ by introducing (\u0000Delta_n(overline{a} ) ) the $n$-th Gram discriminant of ( Z(t) ). We show\u0000that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law [\u0000(-1)^n Delta_n(overline{1}) > 0, ] for any $n in mathbb{Z}$. We further\u0000show that the classical Gram's law ( (-1)^n Z(g_n) >0) can be considered as a\u0000first-order approximation of our corrected law. The second-order approximation\u0000of $Delta_n (overline{a})$ is then shown to be related to shifts of Gram\u0000points along the ( t )-axis. This leads to the discovery of a new, previously\u0000unobserved, repulsion phenomena [ left| Z'(g_n) right| > 4 left| Z(g_n)\u0000right|, ] for bad Gram points $g_n$ whose consecutive neighbours $g_{n pm\u00001}$ are good. Our analysis of the (A)-variation space (mathcal{Z}_N)\u0000introduces a wealth of new results on the zeros of (Z(t)), casting new light\u0000on classical questions such as Gram's law, the Montgomery pair-correlation\u0000conjecture, and the RH, and also unveils previously unknown fundamental\u0000properties.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504905","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 cite{Go}, G"okbac{s} defined a new type of number sequence called Leonardo-Alwyn sequence. In this paper, we consider the generalized Leonardo-Alwyn hybrid numbers and investigate some of their properties. We also give some applications related to the generalized Leonardo-Alwyn hybrid numbers in matrices.
{"title":"Introduction to generalized Leonardo-Alwyn hybrid numbers","authors":"Gamaliel Cerda-Morales","doi":"arxiv-2405.13074","DOIUrl":"https://doi.org/arxiv-2405.13074","url":null,"abstract":"In cite{Go}, G\"okbac{s} defined a new type of number sequence called\u0000Leonardo-Alwyn sequence. In this paper, we consider the generalized\u0000Leonardo-Alwyn hybrid numbers and investigate some of their properties. We also\u0000give some applications related to the generalized Leonardo-Alwyn hybrid numbers\u0000in matrices.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146262","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}
Let $p$ be a large odd prime, let $x=(log p)^{1+varepsilon}$ and let $qllloglog p$ be an integer, where $varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and prime quadratic nonresidues in the arithmetic progression $a+qmll x$, with relatively prime $1leq a
{"title":"Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions","authors":"N. A. Carella","doi":"arxiv-2405.13159","DOIUrl":"https://doi.org/arxiv-2405.13159","url":null,"abstract":"Let $p$ be a large odd prime, let $x=(log p)^{1+varepsilon}$ and let\u0000$qllloglog p$ be an integer, where $varepsilon>0$ is a small number. This\u0000note proves the existence of small prime quadratic residues and prime quadratic\u0000nonresidues in the arithmetic progression $a+qmll x$, with relatively prime\u0000$1leq a<q$, unconditionally. The same results are generalized to small prime\u0000$k$th power residues and nonresidues, where $kmid p-1$ and $kllloglog p$.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146265","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}
N. Anakidze, N. Areshidze, L. -E. Persson, G. Tephnadze
In this paper we present and prove some new results concerning approximation properties of $T$ means with respect to the Vilenkin system in Lebesgue spaces and Lipschitz classes for any $1leq p
在本文中，我们提出并证明了一些新的结果，这些结果是关于任意1美元（leq p
{"title":"Approximation by $T$ means with respect to Vilenkin system in Lebesgue spaces and Lipschitz classes","authors":"N. Anakidze, N. Areshidze, L. -E. Persson, G. Tephnadze","doi":"arxiv-2405.19350","DOIUrl":"https://doi.org/arxiv-2405.19350","url":null,"abstract":"In this paper we present and prove some new results concerning approximation\u0000properties of $T$ means with respect to the Vilenkin system in Lebesgue spaces\u0000and Lipschitz classes for any $1leq p<infty$. As applications, we obtain\u0000extension of some known approximation inequalities.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189204","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 study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Then, we construct some numerical algorithms for these special number systems, as well. Further, we give an introduction for $(s,t)$-Padovan and $(s,t)$-Perrin hyperbolic spinors.
{"title":"Padovan and Perrin Hyperbolic Spinors","authors":"Zehra İşbilir, Işıl Arda Kösal, Murat Tosun","doi":"arxiv-2405.13163","DOIUrl":"https://doi.org/arxiv-2405.13163","url":null,"abstract":"In this study, we intend to bring together Padovan and Perrin number\u0000sequences, which are one of the most popular third-order recurrence sequences,\u0000and hyperbolic spinors, which are used in several disciplines from physics to\u0000mathematics, with the help of the split quaternions. This paper especially\u0000improves the relationship between hyperbolic spinors both a physical and\u0000mathematical concept, and number theory. For this aim, we combine the\u0000hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan\u0000and Perrin quaternions, and we determine two new special recurrence sequences\u0000named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas,\u0000generating functions, exponential generating functions, Poisson generating\u0000functions, and summation formulas. Additionally, we present some matrix and\u0000determinant equations with respect to them. Then, we construct some numerical\u0000algorithms for these special number systems, as well. Further, we give an\u0000introduction for $(s,t)$-Padovan and $(s,t)$-Perrin hyperbolic spinors.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153808","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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