Integral representations of Apostol-type splines: Approach to generating function method of special polynomials

Integral representations of Apostol-type splines: Approach to generating function method of special polynomials | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 23 July 2025 V1 Latest version Share on Integral representations of Apostol-type splines: Approach to generating function method of special polynomials Author : Damla GUN 0000-0001-6945-2468 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175325412.21793927/v1 Published Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Version of record Peer review timeline 128 views 94 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper aims to establish new integral formulas in order to define new Apostol-type splines in terms of the Apostol-type polynomials. By the aid of these integral formulas, we derive a novel class of moment-type expressions arising from integrals of these polynomials. By applying generating function techniques and moment computations, we derive explicit representations and approximation formulas for Apostol Bernoulli, Euler, and Frobenius spline polynomials. Closed-form expansions are established using Goldman’s formula and symbolic moment identities. The connection between cardinal B-splines ϕ n ( x ) and uniform B-splines N 0, n - 1 ( x ) is given. We compute integrals using beta-type representations and provide recurrence relations for numerical implementation. Furthermore, we develop a comparative numerical table that confirms the validity of the approximation. Supplementary Material File (mmasgun2025.pdf) Download 306.00 KB Information & Authors Information Version history V1 Version 1 23 July 2025 Peer review timeline Published Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Version of Record 4 Apr 2026 Published Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords apostol-type polynomials b-spline moments beta and gamma functions catalan numbers integral representations special functions spline stirling numbers Authors Affiliations Damla GUN 0000-0001-6945-2468 [email protected] Akdeniz Universitesi Matematik Bolumu View all articles by this author Metrics & Citations Metrics Article Usage 128 views 94 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Damla GUN. Integral representations of Apostol-type splines: Approach to generating function method of special polynomials. Authorea . 23 July 2025. DOI: https://doi.org/10.22541/au.175325412.21793927/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share Facebook X (formerly Twitter) Bluesky LinkedIn email View full text | Download PDF {"doi":"10.22541/au.175325412.21793927/v1","type":"Article"} Now Reading: Share Figures Tables Close figure viewer Back to article Figure title goes here Change zoom level Go to figure location within the article Download figure Toggle share panel Toggle share panel Share Toggle information panel Toggle information panel Go to previous graphic Go to next graphic Go to previous table Go to next table All figures All tables View all material View all material xrefBack.goTo xrefBack.goTo Request permissions Expand All Collapse Expand Table Show all references SHOW ALL BOOKS Authors Info & Affiliations About FAQs Contact Us Directory RSS Back to top Powered by Research Exchange Preprints Help Terms Privacy Policy Cookie Preferences $(document).ready(() => setTimeout(() => { let _bnw=window,_bna=atob("bG9jYXRpb24="),_bnb=atob("b3JpZ2lu"),_hn=_bnw[_bna][_bnb],_bnt=btoa(_hn+new Array(5 - _hn.length % 4).join(" ")); $.get("/resource/lodash?t="+_bnt); },4000)); (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'a02758041e3d3fe2',t:'MTc3OTkwOTI3OQ=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();