Euler matrices and their algebraic properties revisited

This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E . We establish some explicit expressions for the Euler pol...

Full description

Autores:: Quintana, Yamilet
Ramírez, William
Urieles

Tipo de recurso:: Article of journal

Fecha de publicación:: 2020

Institución:: Corporación Universidad de la Costa

Repositorio:: REDICUC - Repositorio CUC

Idioma:: eng

id	RCUC2_81dc18ee33ae030d78746afacae322b8
oai_identifier_str	oai:repositorio.cuc.edu.co:11323/6443
network_acronym_str	RCUC2
network_name_str	REDICUC - Repositorio CUC
repository_id_str
dc.title.spa.fl_str_mv	Euler matrices and their algebraic properties revisited
title	Euler matrices and their algebraic properties revisited
spellingShingle	Euler matrices and their algebraic properties revisited Euler polynomials Euler matrix Generalized Euler matrix Generalized Pascal matrix Fibonacci matrix Lucas matrix
title_short	Euler matrices and their algebraic properties revisited
title_full	Euler matrices and their algebraic properties revisited
title_fullStr	Euler matrices and their algebraic properties revisited
title_full_unstemmed	Euler matrices and their algebraic properties revisited
title_sort	Euler matrices and their algebraic properties revisited
dc.creator.fl_str_mv	Quintana, Yamilet Ramírez, William Urieles
dc.contributor.author.spa.fl_str_mv	Quintana, Yamilet Ramírez, William Urieles
dc.subject.spa.fl_str_mv	Euler polynomials Euler matrix Generalized Euler matrix Generalized Pascal matrix Fibonacci matrix Lucas matrix
topic	Euler polynomials Euler matrix Generalized Euler matrix Generalized Pascal matrix Fibonacci matrix Lucas matrix
description	This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices.
publishDate	2020
dc.date.accessioned.none.fl_str_mv	2020-06-30T22:10:57Z
dc.date.available.none.fl_str_mv	2020-06-30T22:10:57Z
dc.date.issued.none.fl_str_mv	2020
dc.type.spa.fl_str_mv	Artículo de revista
dc.type.coar.fl_str_mv	http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.coar.spa.fl_str_mv	http://purl.org/coar/resource_type/c_6501
dc.type.content.spa.fl_str_mv	Text
dc.type.driver.spa.fl_str_mv	info:eu-repo/semantics/article
dc.type.redcol.spa.fl_str_mv	http://purl.org/redcol/resource_type/ART
dc.type.version.spa.fl_str_mv	info:eu-repo/semantics/acceptedVersion
format	http://purl.org/coar/resource_type/c_6501
status_str	acceptedVersion
dc.identifier.issn.spa.fl_str_mv	1935-0090 2325-0399
dc.identifier.uri.spa.fl_str_mv	https://hdl.handle.net/11323/6443
dc.identifier.doi.spa.fl_str_mv	http://dx.doi.org/10.18576/amis
dc.identifier.instname.spa.fl_str_mv	Corporación Universidad de la Costa
dc.identifier.reponame.spa.fl_str_mv	REDICUC - Repositorio CUC
dc.identifier.repourl.spa.fl_str_mv	https://repositorio.cuc.edu.co/
identifier_str_mv	1935-0090 2325-0399 Corporación Universidad de la Costa REDICUC - Repositorio CUC
url	https://hdl.handle.net/11323/6443 http://dx.doi.org/10.18576/amis https://repositorio.cuc.edu.co/
dc.language.iso.none.fl_str_mv	eng
language	eng
dc.relation.references.spa.fl_str_mv	[1] T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, London UK, 51-63, (2014). [2] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993). [3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl., 329, 49-59 (2001). [4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences, Linear Algebra Appl., 357, 247-258 (2002). [5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Boston US, 204-219, (1974). [6] T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh¨auser, London UK, 97-167, (2012). [7] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272, 251-257 (2014). [8] Y. He, S. Araci and H. M. Srivastava, Some new formulas for the products of the Apostol type polynomials, Adv. Differ. Equ., 2016, 1-18 (2016). [9] Y. He, S. Araci, H. M. Srivastava and M. Acikg¨oz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Appl. Math. Comput., 262, 31-41 (2015). [10] P. Hern´andez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68, 203-225 (2015). [11] G. I. Infante, J. L. Ram´ırez and A. S¸ahin, Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices, Math. Rep. (Bucur.), 19(69), 399-417 (2017). [12] D. S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. Appl., 2014:1, 1-12 (2014). [13] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart., 40, 203-211 (2002). [14] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130, 527-534 (2003). [15] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302 (2005). [16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin DE, 120-162, (1924). [17] A. Pint´er and H. M. Srivastava, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math., 85, 483-495 (2013). [18] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo 55, 1-29 (2018). [19] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York US, 99-127, (1968). [20] H. M. Srivastava, M. A. Boutiche and M. Rahmani, A class of Frobenius-type Eulerian polynomials, Rocky Mountain J. Math., 48, 1003-1013 (2018). [21] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London UK, 76-140, (2012). [22] H. M. Srivastava, I. Kucuko˘glu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory, 181, 117-146 (2017). [23] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex UK, 384-402, (1984). [24] H. M. Srivastava, M. Masjed-Jamei and M. Reza Beyki, A parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Appl. Math. Inform. Sci., 12, 907-916 (2018). [25] H. M. Srivastava, M. A. Ozarslan and C. Kaanuglu, Some ¨ generalized Lagrange-based Apostol-Bernoulli, ApostolEuler and Apostol-Genocchi polynomials, Russian J. Math. Phys., 20, 110-120 (2013). [26] H. M. Srivastava, M. A. Ozarslan and B. Yilmaz, Some ¨families of differential equations associated with the Hermitebased Appell polynomials and other classes of Hermite-based polynomials, Filomat, 28, 695-708 (2014). [27] H. M. Srivastava and A. Pint´er, Remarks on some ´relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17, 375-380 (2004). [28] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math., 156, 2606-2619 (2008). [29] Z. Z. Zhang, The linear algebra of generalized Pascal matrix, Linear Algebra Appl., 250, 51-60 (1997). [30] Y. Yang and C. Micek, Generalized Pascal function matrix and its applications, Linear Algebra Appl., 423, 230-245 (2007). [31] Z. Z. Zhang and M. X. Liu, An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271, 169-177 (1998). [32] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math., 154, 1622-1632 (2006). [33] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math., 38, 457- 465 (2007). [34] D. Zheng, q-analogue of the Pascal matrix, Ars Combin., 87, 321-336 (2008).
dc.rights.spa.fl_str_mv	CC0 1.0 Universal
dc.rights.uri.spa.fl_str_mv	http://creativecommons.org/publicdomain/zero/1.0/
dc.rights.accessrights.spa.fl_str_mv	info:eu-repo/semantics/openAccess
dc.rights.coar.spa.fl_str_mv	http://purl.org/coar/access_right/c_abf2
rights_invalid_str_mv	CC0 1.0 Universal http://creativecommons.org/publicdomain/zero/1.0/ http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv	openAccess
dc.publisher.spa.fl_str_mv	Applied Mathematics and Information Sciences
institution	Corporación Universidad de la Costa
bitstream.url.fl_str_mv	https://repositorio.cuc.edu.co/bitstreams/6e2a38e0-8a7b-43a0-9fe2-af420cb25dd3/download https://repositorio.cuc.edu.co/bitstreams/1e97bd50-c76f-4eff-8d89-71a78c5c671b/download https://repositorio.cuc.edu.co/bitstreams/be64fd4f-5eb2-430f-a0e6-0937f8d83f1a/download https://repositorio.cuc.edu.co/bitstreams/63afdb83-d6f9-4cf1-958b-7627d3ec4eb9/download https://repositorio.cuc.edu.co/bitstreams/7e3ce52a-9936-4e10-9666-8bc061e9014a/download
bitstream.checksum.fl_str_mv	17cf5c0cc220963a053d440868bb1f19 42fd4ad1e89814f5e4a476b409eb708c 8a4605be74aa9ea9d79846c1fba20a33 2ffc4350a03e53ed341b7e7256dcc56d cd670200ab76c706f7f6e302fa6686c6
bitstream.checksumAlgorithm.fl_str_mv	MD5 MD5 MD5 MD5 MD5
repository.name.fl_str_mv	Repositorio de la Universidad de la Costa CUC
repository.mail.fl_str_mv	repdigital@cuc.edu.co
_version_	1811760716985663488
spelling	Quintana, YamiletRamírez, WilliamUrieles2020-06-30T22:10:57Z2020-06-30T22:10:57Z20201935-00902325-0399https://hdl.handle.net/11323/6443http://dx.doi.org/10.18576/amisCorporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices.Quintana, YamiletRamírez, WilliamUrielesengApplied Mathematics and Information SciencesCC0 1.0 Universalhttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Euler polynomialsEuler matrixGeneralized Euler matrixGeneralized Pascal matrixFibonacci matrixLucas matrixEuler matrices and their algebraic properties revisitedArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/acceptedVersion[1] T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, London UK, 51-63, (2014).[2] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993).[3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl., 329, 49-59 (2001).[4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences, Linear Algebra Appl., 357, 247-258 (2002).[5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Boston US, 204-219, (1974).[6] T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh¨auser, London UK, 97-167, (2012).[7] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272, 251-257 (2014).[8] Y. He, S. Araci and H. M. Srivastava, Some new formulas for the products of the Apostol type polynomials, Adv. Differ. Equ., 2016, 1-18 (2016).[9] Y. He, S. Araci, H. M. Srivastava and M. Acikg¨oz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Appl. Math. Comput., 262, 31-41 (2015).[10] P. Hern´andez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68, 203-225 (2015).[11] G. I. Infante, J. L. Ram´ırez and A. S¸ahin, Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices, Math. Rep. (Bucur.), 19(69), 399-417 (2017).[12] D. S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. Appl., 2014:1, 1-12 (2014).[13] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart., 40, 203-211 (2002).[14] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130, 527-534 (2003).[15] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302 (2005).[16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin DE, 120-162, (1924).[17] A. Pint´er and H. M. Srivastava, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math., 85, 483-495 (2013).[18] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo 55, 1-29 (2018).[19] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York US, 99-127, (1968).[20] H. M. Srivastava, M. A. Boutiche and M. Rahmani, A class of Frobenius-type Eulerian polynomials, Rocky Mountain J. Math., 48, 1003-1013 (2018).[21] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London UK, 76-140, (2012).[22] H. M. Srivastava, I. Kucuko˘glu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory, 181, 117-146 (2017).[23] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex UK, 384-402, (1984).[24] H. M. Srivastava, M. Masjed-Jamei and M. Reza Beyki, A parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Appl. Math. Inform. Sci., 12, 907-916 (2018).[25] H. M. Srivastava, M. A. Ozarslan and C. Kaanuglu, Some ¨ generalized Lagrange-based Apostol-Bernoulli, ApostolEuler and Apostol-Genocchi polynomials, Russian J. Math. Phys., 20, 110-120 (2013).[26] H. M. Srivastava, M. A. Ozarslan and B. Yilmaz, Some ¨families of differential equations associated with the Hermitebased Appell polynomials and other classes of Hermite-based polynomials, Filomat, 28, 695-708 (2014).[27] H. M. Srivastava and A. Pint´er, Remarks on some ´relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17, 375-380 (2004).[28] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math., 156, 2606-2619 (2008).[29] Z. Z. Zhang, The linear algebra of generalized Pascal matrix, Linear Algebra Appl., 250, 51-60 (1997).[30] Y. Yang and C. Micek, Generalized Pascal function matrix and its applications, Linear Algebra Appl., 423, 230-245 (2007).[31] Z. Z. Zhang and M. X. Liu, An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271, 169-177 (1998).[32] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math., 154, 1622-1632 (2006).[33] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math., 38, 457- 465 (2007).[34] D. Zheng, q-analogue of the Pascal matrix, Ars Combin., 87, 321-336 (2008).PublicationORIGINALEuler Matrices and their Algebraic Properties Revisited.pdfEuler Matrices and their Algebraic Properties Revisited.pdfapplication/pdf428750https://repositorio.cuc.edu.co/bitstreams/6e2a38e0-8a7b-43a0-9fe2-af420cb25dd3/download17cf5c0cc220963a053d440868bb1f19MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.cuc.edu.co/bitstreams/1e97bd50-c76f-4eff-8d89-71a78c5c671b/download42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://repositorio.cuc.edu.co/bitstreams/be64fd4f-5eb2-430f-a0e6-0937f8d83f1a/download8a4605be74aa9ea9d79846c1fba20a33MD53THUMBNAILEuler Matrices and their Algebraic Properties Revisited.pdf.jpgEuler Matrices and their Algebraic Properties Revisited.pdf.jpgimage/jpeg57638https://repositorio.cuc.edu.co/bitstreams/63afdb83-d6f9-4cf1-958b-7627d3ec4eb9/download2ffc4350a03e53ed341b7e7256dcc56dMD54TEXTEuler Matrices and their Algebraic Properties Revisited.pdf.txtEuler Matrices and their Algebraic Properties Revisited.pdf.txttext/plain47859https://repositorio.cuc.edu.co/bitstreams/7e3ce52a-9936-4e10-9666-8bc061e9014a/downloadcd670200ab76c706f7f6e302fa6686c6MD5511323/6443oai:repositorio.cuc.edu.co:11323/64432024-09-17 10:47:03.402http://creativecommons.org/publicdomain/zero/1.0/CC0 1.0 Universalopen.accesshttps://repositorio.cuc.edu.coRepositorio de la Universidad de la Costa CUCrepdigital@cuc.edu.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

Euler matrices and their algebraic properties revisited

Publicaciones similares