{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T01:22:11Z","timestamp":1775611331592,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,15]],"date-time":"2019-04-15T00:00:00Z","timestamp":1555286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.<\/jats:p>","DOI":"10.3390\/sym11040538","type":"journal-article","created":{"date-parts":[[2019,4,15]],"date-time":"2019-04-15T11:15:58Z","timestamp":1555326958000},"page":"538","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["A Note on the Truncated-Exponential Based Apostol-Type Polynomials"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"H. M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3950-6864","authenticated-orcid":false,"given":"Serkan","family":"Araci","sequence":"additional","affiliation":[{"name":"Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Waseem A.","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Integral University, Lucknow 226026, Uttar Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mehmet","family":"Acikg\u00f6z","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Gaziantep University, TR-27310 Gaziantep, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02392231","article-title":"The poweroid, an extension of the mathematical notion of power","volume":"73","author":"Steffensen","year":"1941","journal-title":"Acta Math."},{"key":"ref_2","unstructured":"Cocolicchio, D., Dattoli, G., and Srivastava, H.M. (2000). Hermite-Bessel and Laguerre-Bessel-functions: A by-product of the monomiality principle. Advanced Special Functions and Applications, Proceedings of the First Melfi School on Advanced Topics in Mathematics and Physics, Melfi, Italy, 9\u201312 May 1999, Aracne Editrice."},{"key":"ref_3","unstructured":"Andrews, L.C. (1985). Special Functions for Engineers and Mathematicians, Macmillan Company."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1016\/S0096-3003(01)00310-1","article-title":"A note on truncated polynomials","volume":"134","author":"Dattoli","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_5","first-page":"169","article-title":"A class of Bessel summation formulas and associated operational methods","volume":"7","author":"Dattoli","year":"2004","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1016\/j.jmaa.2014.04.028","article-title":"On a new family related to truncated exponential and Sheffer polynomials","volume":"418","author":"Khan","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1007\/s00009-015-0610-7","article-title":"Operational methods and truncated exponential-based Mittag-Leffler polynomials","volume":"13","author":"Yasmin","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2140\/pjm.1951.1.161","article-title":"On the Lerch zeta function","volume":"1","author":"Apostol","year":"1951","journal-title":"Pac. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"S\u00e1ndor, J., and Crsci, B. (2004). Handbook of Number Theory, Kluwer Academic Publishers.","DOI":"10.1007\/1-4020-2547-5"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2001). Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-9672-5"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cattani, C., Srivastava, H.M., and Yang, X.-J. (2015). Fractional Derivative of the Riemann Zeta function. Fractional Dynamics, Emerging Science Publishers (De Gruyter Open).","DOI":"10.1515\/9783110472097-022"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1847-2013-361","article-title":"Some relations involving generalized Hurwitz-Lerch zeta function obtained by means of fractional derivatives with applications to Apostol-type polynomials","volume":"2013","author":"Gaboury","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1016\/S0096-3003(03)00746-X","article-title":"Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations","volume":"154","author":"Lin","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1017\/S0305004100004412","article-title":"Some formulas for the Bernoulli and Euler polynomials at rational arguments","volume":"129","author":"Srivastava","year":"2000","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_15","first-page":"020063","article-title":"A functional equation for the Riemann zeta fractional derivative","volume":"1798","author":"Guariglia","year":"2017","journal-title":"AIP Conf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.camwa.2005.04.018","article-title":"Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials","volume":"51","author":"Luo","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.jmaa.2005.01.020","article-title":"Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials","volume":"308","author":"Luo","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2193","DOI":"10.1090\/S0025-5718-09-02230-3","article-title":"Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials","volume":"78","author":"Luo","year":"2009","journal-title":"Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2993","DOI":"10.1016\/j.camwa.2008.07.038","article-title":"Several identities for the generalized Apostol-Bernoulli polynomials","volume":"56","author":"Zhang","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"336","DOI":"10.2298\/AADM0902336L","article-title":"Some formulas for the Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments","volume":"3","author":"Luo","year":"2009","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1847-2014-155","article-title":"Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials","volume":"2014","author":"He","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-016-1014-0","article-title":"Some new formulas for the products of the Apostol type polynomials","volume":"2016","author":"He","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.amc.2015.03.132","article-title":"Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials","volume":"262","author":"He","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_24","first-page":"1","article-title":"Fourier expansions and integral representations for the Genocchi polynomials","volume":"12","author":"Luo","year":"2009","journal-title":"J. Integer Seq."},{"key":"ref_25","first-page":"113","article-title":"q-Extension for the Apostol-Genocchi polynomials","volume":"17","author":"Luo","year":"2009","journal-title":"Gen. Math."},{"key":"ref_26","first-page":"291","article-title":"Extensions for the Genocchi polynomials and their Fourier expansions and integral representations","volume":"48","author":"Luo","year":"2011","journal-title":"Osaka J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5702","DOI":"10.1016\/j.amc.2010.12.048","article-title":"Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind","volume":"217","author":"Luo","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"\u00d6zden, H. (2010). Unification of generating functions of the Bernoulli, Euler and Genocchi numbers and polynomials. AIP Conf. Proc.","DOI":"10.1063\/1.3497848"},{"key":"ref_29","first-page":"349","article-title":"Generating function of the unified representation of the Bernoulli, Euler and Genocchi polynomials of higher order","volume":"1389","year":"2011","journal-title":"AIP Conf. Proc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2779","DOI":"10.1016\/j.camwa.2010.09.031","article-title":"A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials","volume":"60","author":"Simsek","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_31","first-page":"1","article-title":"Hermite-Based unified Apostol-Bernoulli, Euler and Genocchi polynomials","volume":"2013","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_32","first-page":"2452","article-title":"Unified Apostol-Bernoulli, Euler and Genocchi polynomials","volume":"6","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.jmaa.2008.02.052","article-title":"Implicit summation formula for Hermite and related polynomials","volume":"344","author":"Khan","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","first-page":"113","article-title":"Some implicit summation formulas and symmetric identities for the generalized Hermite-based polynomials","volume":"39","author":"Pathan","year":"2014","journal-title":"Acta Univ. Apulensis."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1007\/s00009-014-0423-0","article-title":"Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials","volume":"12","author":"Pathan","year":"2015","journal-title":"Mediterr. J. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1007\/s00009-015-0551-1","article-title":"A new class of generalized polynomials associated with Hermite and Euler polynomials","volume":"13","author":"Pathan","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_37","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Wiley and Sons."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"597","DOI":"10.5666\/KMJ.2015.55.3.597","article-title":"Some properties of the generalized Apostol type Hermite-Based polynomials","volume":"55","author":"Khan","year":"2015","journal-title":"Kyungpook Math. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/538\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:45:27Z","timestamp":1760186727000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/538"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,15]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040538"],"URL":"https:\/\/doi.org\/10.3390\/sym11040538","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,15]]}}}