Generalized apostol-type polynomial matrix and its algebraic properties

The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α](x;c,a;λ;µ;ν) and the generalized Apos-tol-type matrix W [m−1,α](c,a;λ;µ;ν). Using some properties of the generalized Apostol-type polynomials and numbers, we deduce a product formula for W [m−1,α](x;c,a;λ;...

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Autores:
Quintana, Yamilet
Ramírez, William
Urieles Guerrero, Alejandro
Tipo de recurso:
Article of journal
Fecha de publicación:
2019
Institución:
Corporación Universidad de la Costa
Repositorio:
REDICUC - Repositorio CUC
Idioma:
eng
OAI Identifier:
oai:repositorio.cuc.edu.co:11323/5649
Acceso en línea:
https://hdl.handle.net/11323/5649
https://repositorio.cuc.edu.co/
Palabra clave:
Generalized Apostol-type polynomials
Generalized Apostol-type matrix
Admissible generalized Apostol-type matrix
Generalized Pascal matrix
Generalized Fibonacci matrix
Lucas matrix
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openAccess
License
CC0 1.0 Universal
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oai_identifier_str oai:repositorio.cuc.edu.co:11323/5649
network_acronym_str RCUC2
network_name_str REDICUC - Repositorio CUC
repository_id_str
dc.title.spa.fl_str_mv Generalized apostol-type polynomial matrix and its algebraic properties
title Generalized apostol-type polynomial matrix and its algebraic properties
spellingShingle Generalized apostol-type polynomial matrix and its algebraic properties
Generalized Apostol-type polynomials
Generalized Apostol-type matrix
Admissible generalized Apostol-type matrix
Generalized Pascal matrix
Generalized Fibonacci matrix
Lucas matrix
title_short Generalized apostol-type polynomial matrix and its algebraic properties
title_full Generalized apostol-type polynomial matrix and its algebraic properties
title_fullStr Generalized apostol-type polynomial matrix and its algebraic properties
title_full_unstemmed Generalized apostol-type polynomial matrix and its algebraic properties
title_sort Generalized apostol-type polynomial matrix and its algebraic properties
dc.creator.fl_str_mv Quintana, Yamilet
Ramírez, William
Urieles Guerrero, Alejandro
dc.contributor.author.spa.fl_str_mv Quintana, Yamilet
Ramírez, William
Urieles Guerrero, Alejandro
dc.subject.spa.fl_str_mv Generalized Apostol-type polynomials
Generalized Apostol-type matrix
Admissible generalized Apostol-type matrix
Generalized Pascal matrix
Generalized Fibonacci matrix
Lucas matrix
topic Generalized Apostol-type polynomials
Generalized Apostol-type matrix
Admissible generalized Apostol-type matrix
Generalized Pascal matrix
Generalized Fibonacci matrix
Lucas matrix
description The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α](x;c,a;λ;µ;ν) and the generalized Apos-tol-type matrix W [m−1,α](c,a;λ;µ;ν). Using some properties of the generalized Apostol-type polynomials and numbers, we deduce a product formula for W [m−1,α](x;c,a;λ;µ;ν) and provide some factorizations of the Apostol-type polynomial matrix W [m−1](x;c,a;λ;µ;ν), involving the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively. AMS 2010 Subject Classification: 11B68, 11B83, 11C08, 11B39, 33B99.
publishDate 2019
dc.date.accessioned.none.fl_str_mv 2019-11-13T20:14:06Z
dc.date.available.none.fl_str_mv 2019-11-13T20:14:06Z
dc.date.issued.none.fl_str_mv 2019
dc.type.spa.fl_str_mv Artículo de revista
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dc.identifier.issn.spa.fl_str_mv 2285-3898
1582-3067
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dc.identifier.instname.spa.fl_str_mv Corporación Universidad de la Costa
dc.identifier.reponame.spa.fl_str_mv REDICUC - Repositorio CUC
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identifier_str_mv 2285-3898
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Corporación Universidad de la Costa
REDICUC - Repositorio CUC
url https://hdl.handle.net/11323/5649
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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 Monogr. Math., Springer, Tokyo, Heidelberg, New York, Dordrecht and London, 2014. [2] G.S. Call and D.J. Velleman, Pascal’s matrices. Amer. Math. Monthly 100 (1993), 372–376. [3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix. Linear Algebra Appl. 329 (2001), 49–59. [4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences. Linear Algebra Appl. 357 (2002), 247–258. [5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions. D. Reidel Publishing Co., Dordrecht and Boston, 1974. (Translated from French by J.W. Nienhuys). [6] T. Ernst, A Comprehensive Treatment of q-Calculus. Birkha¨user, Springer, Basel, Heidelberg, New York, Dordrecht and London, 2012. [7] P. Herna´ndez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostoltype polynomials. Results Math. 68 (2015), 203–225. [8] 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 (2017), 4, 399–417. [9] D.S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus. J. Inequal. Appl. 2014 (2014), Paper No. 1, 12 p. [10] G.-Y. Lee and J.-S. Kim, The linear algebra of the k-Fibonacci matrix. Linear Algebra Appl. 373 (2003), 75–87. [11] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. Fibonacci Quart. 40 (2002), 3, 203–211. [12] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers. Discrete Appl. Math. 130 (2003), 3, 527–534. [13] B. Kurt, Some relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. Turkish J. Anal. Number Theory 1 (2013), 1, 54–58. [14] D.-Q. Lu and Q.-M. Luo, Some properties of the generalized Apostol-type polynomials. Bound. Value Probl. 2013 (2013), Paper No. 64, 13 p. [15] Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308 (2005), 1, 290–302. [16] Q.-M. Luo and H.M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 51 (2006), 631–642. [17] Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217 (2011), 5702–5728. [18] H. Ozden, Y. Simsek and H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernolli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (2010), 2779–2787. [19] Y. Quintana, W. Ram´ırez and A. Urieles, Euler matrices and their algebraic properties revisited. Submitted. arXiv:1811.01455 [math.NT]. [20] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m. Calcolo 55 (2018), 3, Paper No. 30, 29 p. [21] J. Riordan, Combinatorial Identities. Wiley, New York, London and Sydney, 1968. [22] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions. Springer, Dordrecht, Netherlands, 2001. [23] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, London, 2012. [24] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices. Discrete Appl. Math. 156 (2008), 14, 2606–2619. [25] W. Wang and T. Wang, Matrices related to the Bell polynomials. Linear Algebra Appl. 422 (2007), 139–154. [26] Z.Z. Zhang, The linear algebra of generalized Pascal matrix. Linear Algebra Appl. 250 (1997), 51–60. [27] Z.Z. Zhang and M.X. Liu, An extension of generalized Pascal matrix and its algebraic properties. Linear Algebra Appl. 271 (1998), 169–177. [28] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties. Discrete Appl. Math. 154 (2006), 1622–1632. [29] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities. Indian J. Pure Appl. Math. 38 (2007), 5, 457–465.
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spelling Quintana, YamiletRamírez, WilliamUrieles Guerrero, Alejandro2019-11-13T20:14:06Z2019-11-13T20:14:06Z20192285-38981582-3067https://hdl.handle.net/11323/5649Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α](x;c,a;λ;µ;ν) and the generalized Apos-tol-type matrix W [m−1,α](c,a;λ;µ;ν). Using some properties of the generalized Apostol-type polynomials and numbers, we deduce a product formula for W [m−1,α](x;c,a;λ;µ;ν) and provide some factorizations of the Apostol-type polynomial matrix W [m−1](x;c,a;λ;µ;ν), involving the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively. AMS 2010 Subject Classification: 11B68, 11B83, 11C08, 11B39, 33B99.Quintana, YamiletRamírez, WilliamUrieles Guerrero, Alejandro-will be generated-orcid-0000-0002-7186-0898-600engMathematical ReportsCC0 1.0 Universalhttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Generalized Apostol-type polynomialsGeneralized Apostol-type matrixAdmissible generalized Apostol-type matrixGeneralized Pascal matrixGeneralized Fibonacci matrixLucas matrixGeneralized apostol-type polynomial matrix and its algebraic propertiesArtículo 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 Monogr. Math., Springer, Tokyo, Heidelberg, New York, Dordrecht and London, 2014. [2] G.S. Call and D.J. Velleman, Pascal’s matrices. Amer. Math. Monthly 100 (1993), 372–376. [3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix. Linear Algebra Appl. 329 (2001), 49–59. [4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences. Linear Algebra Appl. 357 (2002), 247–258. [5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions. D. Reidel Publishing Co., Dordrecht and Boston, 1974. (Translated from French by J.W. Nienhuys). [6] T. Ernst, A Comprehensive Treatment of q-Calculus. Birkha¨user, Springer, Basel, Heidelberg, New York, Dordrecht and London, 2012. [7] P. Herna´ndez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostoltype polynomials. Results Math. 68 (2015), 203–225. [8] 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 (2017), 4, 399–417. [9] D.S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus. J. Inequal. Appl. 2014 (2014), Paper No. 1, 12 p. [10] G.-Y. Lee and J.-S. Kim, The linear algebra of the k-Fibonacci matrix. Linear Algebra Appl. 373 (2003), 75–87. [11] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. Fibonacci Quart. 40 (2002), 3, 203–211. [12] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers. Discrete Appl. Math. 130 (2003), 3, 527–534. [13] B. Kurt, Some relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. Turkish J. Anal. Number Theory 1 (2013), 1, 54–58. [14] D.-Q. Lu and Q.-M. Luo, Some properties of the generalized Apostol-type polynomials. Bound. Value Probl. 2013 (2013), Paper No. 64, 13 p. [15] Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308 (2005), 1, 290–302. [16] Q.-M. Luo and H.M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 51 (2006), 631–642. [17] Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217 (2011), 5702–5728. [18] H. Ozden, Y. Simsek and H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernolli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (2010), 2779–2787. [19] Y. Quintana, W. Ram´ırez and A. Urieles, Euler matrices and their algebraic properties revisited. Submitted. arXiv:1811.01455 [math.NT]. [20] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m. Calcolo 55 (2018), 3, Paper No. 30, 29 p. [21] J. Riordan, Combinatorial Identities. Wiley, New York, London and Sydney, 1968. [22] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions. Springer, Dordrecht, Netherlands, 2001. [23] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, London, 2012. [24] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices. Discrete Appl. Math. 156 (2008), 14, 2606–2619. [25] W. Wang and T. Wang, Matrices related to the Bell polynomials. Linear Algebra Appl. 422 (2007), 139–154. [26] Z.Z. Zhang, The linear algebra of generalized Pascal matrix. Linear Algebra Appl. 250 (1997), 51–60. [27] Z.Z. Zhang and M.X. Liu, An extension of generalized Pascal matrix and its algebraic properties. Linear Algebra Appl. 271 (1998), 169–177. [28] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties. Discrete Appl. Math. 154 (2006), 1622–1632. [29] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities. Indian J. Pure Appl. Math. 38 (2007), 5, 457–465.PublicationORIGINALGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdfGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdfapplication/pdf633164https://repositorio.cuc.edu.co/bitstreams/bca4705e-69cd-4be0-9d3c-46682f8baa15/downloadbb0e422e6bc29e4969e019c933309450MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.cuc.edu.co/bitstreams/2568af4b-24b5-4dad-ad17-99d6390a1fb5/download42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://repositorio.cuc.edu.co/bitstreams/0951e295-b621-4092-a56c-431dda6285a3/download8a4605be74aa9ea9d79846c1fba20a33MD53THUMBNAILGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdf.jpgGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdf.jpgimage/jpeg37367https://repositorio.cuc.edu.co/bitstreams/60f373a5-fe24-44b2-a608-477a69504e94/download7d97af13440247e2bf82b7cde80a9265MD55TEXTGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdf.txtGENERALIZED APOSTOL-TYPE POLYNOMIAL MATRIX AND ITS ALGEBRAIC PROPERTIES.pdf.txttext/plain30430https://repositorio.cuc.edu.co/bitstreams/5d2c5187-cedb-4b9f-98b1-4646ba755379/download2d80c5f8f6b3ffcdc40c2d367e40fdb5MD5611323/5649oai:repositorio.cuc.edu.co:11323/56492024-09-17 14:13:47.617http://creativecommons.org/publicdomain/zero/1.0/CC0 1.0 Universalopen.accesshttps://repositorio.cuc.edu.coRepositorio de la Universidad de la Costa CUCrepdigital@cuc.edu.coTk9URTogUExBQ0UgWU9VUiBPV04gTElDRU5TRSBIRVJFClRoaXMgc2FtcGxlIGxpY2Vuc2UgaXMgcHJvdmlkZWQgZm9yIGluZm9ybWF0aW9uYWwgcHVycG9zZXMgb25seS4KCk5PTi1FWENMVVNJVkUgRElTVFJJQlVUSU9OIExJQ0VOU0UKCkJ5IHNpZ25pbmcgYW5kIHN1Ym1pdHRpbmcgdGhpcyBsaWNlbnNlLCB5b3UgKHRoZSBhdXRob3Iocykgb3IgY29weXJpZ2h0Cm93bmVyKSBncmFudHMgdG8gRFNwYWNlIFVuaXZlcnNpdHkgKERTVSkgdGhlIG5vbi1leGNsdXNpdmUgcmlnaHQgdG8gcmVwcm9kdWNlLAp0cmFuc2xhdGUgKGFzIGRlZmluZWQgYmVsb3cpLCBhbmQvb3IgZGlzdHJpYnV0ZSB5b3VyIHN1Ym1pc3Npb24gKGluY2x1ZGluZwp0aGUgYWJzdHJhY3QpIHdvcmxkd2lkZSBpbiBwcmludCBhbmQgZWxlY3Ryb25pYyBmb3JtYXQgYW5kIGluIGFueSBtZWRpdW0sCmluY2x1ZGluZyBidXQgbm90IGxpbWl0ZWQgdG8gYXVkaW8gb3IgdmlkZW8uCgpZb3UgYWdyZWUgdGhhdCBEU1UgbWF5LCB3aXRob3V0IGNoYW5naW5nIHRoZSBjb250ZW50LCB0cmFuc2xhdGUgdGhlCnN1Ym1pc3Npb24gdG8gYW55IG1lZGl1bSBvciBmb3JtYXQgZm9yIHRoZSBwdXJwb3NlIG9mIHByZXNlcnZhdGlvbi4KCllvdSBhbHNvIGFncmVlIHRoYXQgRFNVIG1heSBrZWVwIG1vcmUgdGhhbiBvbmUgY29weSBvZiB0aGlzIHN1Ym1pc3Npb24gZm9yCnB1cnBvc2VzIG9mIHNlY3VyaXR5LCBiYWNrLXVwIGFuZCBwcmVzZXJ2YXRpb24uCgpZb3UgcmVwcmVzZW50IHRoYXQgdGhlIHN1Ym1pc3Npb24gaXMgeW91ciBvcmlnaW5hbCB3b3JrLCBhbmQgdGhhdCB5b3UgaGF2ZQp0aGUgcmlnaHQgdG8gZ3JhbnQgdGhlIHJpZ2h0cyBjb250YWluZWQgaW4gdGhpcyBsaWNlbnNlLiBZb3UgYWxzbyByZXByZXNlbnQKdGhhdCB5b3VyIHN1Ym1pc3Npb24gZG9lcyBub3QsIHRvIHRoZSBiZXN0IG9mIHlvdXIga25vd2xlZGdlLCBpbmZyaW5nZSB1cG9uCmFueW9uZSdzIGNvcHlyaWdodC4KCklmIHRoZSBzdWJtaXNzaW9uIGNvbnRhaW5zIG1hdGVyaWFsIGZvciB3aGljaCB5b3UgZG8gbm90IGhvbGQgY29weXJpZ2h0LAp5b3UgcmVwcmVzZW50IHRoYXQgeW91IGhhdmUgb2J0YWluZWQgdGhlIHVucmVzdHJpY3RlZCBwZXJtaXNzaW9uIG9mIHRoZQpjb3B5cmlnaHQgb3duZXIgdG8gZ3JhbnQgRFNVIHRoZSByaWdodHMgcmVxdWlyZWQgYnkgdGhpcyBsaWNlbnNlLCBhbmQgdGhhdApzdWNoIHRoaXJkLXBhcnR5IG93bmVkIG1hdGVyaWFsIGlzIGNsZWFybHkgaWRlbnRpZmllZCBhbmQgYWNrbm93bGVkZ2VkCndpdGhpbiB0aGUgdGV4dCBvciBjb250ZW50IG9mIHRoZSBzdWJtaXNzaW9uLgoKSUYgVEhFIFNVQk1JU1NJT04gSVMgQkFTRUQgVVBPTiBXT1JLIFRIQVQgSEFTIEJFRU4gU1BPTlNPUkVEIE9SIFNVUFBPUlRFRApCWSBBTiBBR0VOQ1kgT1IgT1JHQU5JWkFUSU9OIE9USEVSIFRIQU4gRFNVLCBZT1UgUkVQUkVTRU5UIFRIQVQgWU9VIEhBVkUKRlVMRklMTEVEIEFOWSBSSUdIVCBPRiBSRVZJRVcgT1IgT1RIRVIgT0JMSUdBVElPTlMgUkVRVUlSRUQgQlkgU1VDSApDT05UUkFDVCBPUiBBR1JFRU1FTlQuCgpEU1Ugd2lsbCBjbGVhcmx5IGlkZW50aWZ5IHlvdXIgbmFtZShzKSBhcyB0aGUgYXV0aG9yKHMpIG9yIG93bmVyKHMpIG9mIHRoZQpzdWJtaXNzaW9uLCBhbmQgd2lsbCBub3QgbWFrZSBhbnkgYWx0ZXJhdGlvbiwgb3RoZXIgdGhhbiBhcyBhbGxvd2VkIGJ5IHRoaXMKbGljZW5zZSwgdG8geW91ciBzdWJtaXNzaW9uLgo=

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