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
OAI Identifier:
oai:repositorio.cuc.edu.co:11323/6443
Acceso en línea:
https://hdl.handle.net/11323/6443
http://dx.doi.org/10.18576/amis
https://repositorio.cuc.edu.co/
Palabra clave:
Euler polynomials
Euler matrix
Generalized Euler matrix
Generalized Pascal matrix
Fibonacci matrix
Lucas matrix
Rights
openAccess
License
CC0 1.0 Universal
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
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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).
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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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