Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobeni...

Full description

Autores:
Urieles Guerrero, Alejandro
Ramírez, William
Ortega, María José
Bedoya, Daniel
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:11323/7140
Acceso en línea:
https://hdl.handle.net/11323/7140
https://doi.org/10.1186/s13662-020-02988-0
https://repositorio.cuc.edu.co/
Palabra clave:
Generalized apostol frobenius–euler polynomials
Hurwitz zeta function
Fourier expansion
Generalized apostol frobennius–euler numbers
Rights
openAccess
License
CC0 1.0 Universal
id RCUC2_888ba84b6564ecc1854f4443be3f1b46
oai_identifier_str oai:repositorio.cuc.edu.co:11323/7140
network_acronym_str RCUC2
network_name_str REDICUC - Repositorio CUC
repository_id_str
dc.title.spa.fl_str_mv Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
title Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
spellingShingle Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
Generalized apostol frobenius–euler polynomials
Hurwitz zeta function
Fourier expansion
Generalized apostol frobennius–euler numbers
title_short Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
title_full Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
title_fullStr Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
title_full_unstemmed Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
title_sort Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials
dc.creator.fl_str_mv Urieles Guerrero, Alejandro
Ramírez, William
Ortega, María José
Bedoya, Daniel
dc.contributor.author.spa.fl_str_mv Urieles Guerrero, Alejandro
Ramírez, William
Ortega, María José
Bedoya, Daniel
dc.subject.spa.fl_str_mv Generalized apostol frobenius–euler polynomials
Hurwitz zeta function
Fourier expansion
Generalized apostol frobennius–euler numbers
topic Generalized apostol frobenius–euler polynomials
Hurwitz zeta function
Fourier expansion
Generalized apostol frobennius–euler numbers
description The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.
publishDate 2020
dc.date.accessioned.none.fl_str_mv 2020-10-14T23:23:10Z
dc.date.available.none.fl_str_mv 2020-10-14T23:23:10Z
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 1687-1839
1687-1847
dc.identifier.uri.spa.fl_str_mv https://hdl.handle.net/11323/7140
dc.identifier.doi.spa.fl_str_mv https://doi.org/10.1186/s13662-020-02988-0
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 1687-1839
1687-1847
Corporación Universidad de la Costa
REDICUC - Repositorio CUC
url https://hdl.handle.net/11323/7140
https://doi.org/10.1186/s13662-020-02988-0
https://repositorio.cuc.edu.co/
dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.references.spa.fl_str_mv 1. Alkan, M., Simsek, Y.: Generating function for q-Eulerian polynomials and their decomposition and applications. Fixed Point Theory Appl. 2013(72), 1 (2013). https://doi.org/10.1186/1687-1812-2013-72
2. Araci, S., Acikgoz, M.: Construction of Fourier expansion of Apostol Frobenius–Euler polynomials and its applications. Adv. Differ. Equ. 2018, 67 (2018). https://doi.org/10.1186/s13662-018-1526-x
3. Bayad, A.: Fourier expansion for Apostol Bernoulli, Apostol Euler and Apostol Genocchi polynomials. Math. Comput. 80, 2219–2221 (2011). https://doi.org/10.1090/S0025-5718-2011-02476-2
4. Bayad, A., Kim, T.: Identities for Apostol-type Frobenius–Euler polynomiasl resulting from the study of a nonlinear operator. Russ. J. Math. Phys. 23, 164–171 (2016). https://doi.org/10.1134/S1061920816020023
5. Cangul, I.N., Cevik, A.S., Simsek, Y.: Generalization of q-Apostol-type Eulerian numbers and polynomials, and their interpolation functions. Adv. Stud. Contemp. Math. 25(2), 211–220 (2015)
6. Carlitz, L.: Eulerian numbers and polynomials. Math. Mag. 32, 247–260 (1959). https://doi.org/10.2307/3029225
7. Conway, J.B.: Functions of One Complex Variables. Springer, Berlin (1978)
8. Cristina, B., Roberto, B.: Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli and Apostol–Euler polynomialsv. Adv. Differ. Equ. 2020, 346 (2020). https://doi.org/10.1186/s13662-020-02802-x
9. Follan, G.: Fourier Analysis and Its Applications (1992)
10. Kim, T.: An identity of the symmetry for the Frobenius–Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp. Rocky Mt. J. Math. 41, 239–247 (2011)
11. Kucukoglu, I., Simsek, Y.: Identities and relations on the q-Apostol type Frobenius–Euler numbers and polynomials. J. Korean Math. Soc. 56(1), 265–284 (2019). https://doi.org/10.4134/JKMS.j180185
12. Kucukoglu, I., Simsek, Y., Srivastava, H.M.: A new family of Lerch-type zeta functions interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials. Quaest. Math. 42 465–478 (2019). https://doi.org/10.2989/16073606.2018.1459925
13. Kurt, B., Simsek, Y.: On the generalized Apostol-type Frobenius–Euler polynomials. Adv. Differ. Equ. 2013, 1 (2013). https://doi.org/10.1186/1687-1847-2013-1
14. Luo, Q.: Fourier expansion and integral representations for the Apostol Bernoulli and Apostol Euler polynomials. Math. Comput. 78, 2193–2208 (2009)
15. Luo, Q.-M.: Extensions of the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 48, 291–309 (2011)
16. Quintana, Y., Ramírez, W., Urieles, A.: Euler matrices and their algebraic properties revisited. Appl. Math. Inf. Sci. 14(4), 583–596 (2020). https://doi.org/10.18576/amis/140407
17. Ramírez, W., Ortega, M., Urieles, A.: New generalized Apostol Frobenius–Euler polynomials and their matrix approach. Kragujev. J. Math. 45(3), 393–407 (2021)
18. Simsek, Y.: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. Fixed Point Theory Appl. 2013(87), 1 (2013). https://doi.org/10.1186/1687-1812-2013-87
19. Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)
20. Srivastava, H.M., Kurt, B., Simsek, Y.: Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms Spec. Funct. 23(12), 919–938 (2012). https://doi.org/10.1080/10652469.2011.643627
21. Yilmaz, S.: Generating functions for q-Apostol type Frobenius–Euler numbers and polynomials. Axioms 1(3), 395–403 (2012). https://doi.org/10.3390/axioms1030395
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 Corporación Universidad de la Costa
dc.source.spa.fl_str_mv Advances in Difference Equations
institution Corporación Universidad de la Costa
dc.source.url.spa.fl_str_mv https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02988-0
bitstream.url.fl_str_mv https://repositorio.cuc.edu.co/bitstreams/18787330-7b33-4398-9d11-985d823e13f5/download
https://repositorio.cuc.edu.co/bitstreams/3336be6f-94ed-49bc-ba14-02e5cad6aa3b/download
https://repositorio.cuc.edu.co/bitstreams/9429157d-d872-4594-9956-f7d176bd7f36/download
https://repositorio.cuc.edu.co/bitstreams/f0eb1ff1-fa37-470a-83ae-266026ddcd97/download
https://repositorio.cuc.edu.co/bitstreams/617633e6-4e8d-4314-9988-1b68cf773dd6/download
https://repositorio.cuc.edu.co/bitstreams/d16090c9-130c-4685-a48c-e20555853e2f/download
bitstream.checksum.fl_str_mv 7bcff39f2a974e7da3115aabdfd2d9df
42fd4ad1e89814f5e4a476b409eb708c
e30e9215131d99561d40d6b0abbe9bad
4f22d98e515755a505c451b1a911d75a
4f22d98e515755a505c451b1a911d75a
0cd99bc72be91e085b55497e4dae8e29
bitstream.checksumAlgorithm.fl_str_mv MD5
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_ 1811760697253560320
spelling Urieles Guerrero, AlejandroRamírez, WilliamOrtega, María JoséBedoya, Daniel2020-10-14T23:23:10Z2020-10-14T23:23:10Z20201687-18391687-1847https://hdl.handle.net/11323/7140https://doi.org/10.1186/s13662-020-02988-0Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.Urieles Guerrero, Alejandro-will be generated-orcid-0000-0002-7186-0898-600Ramírez, WilliamOrtega, María JoséBedoya, DanielengCorporación Universidad de la CostaCC0 1.0 Universalhttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Advances in Difference Equationshttps://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02988-0Generalized apostol frobenius–euler polynomialsHurwitz zeta functionFourier expansionGeneralized apostol frobennius–euler numbersFourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomialsArtículo 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/acceptedVersion1. Alkan, M., Simsek, Y.: Generating function for q-Eulerian polynomials and their decomposition and applications. Fixed Point Theory Appl. 2013(72), 1 (2013). https://doi.org/10.1186/1687-1812-2013-722. Araci, S., Acikgoz, M.: Construction of Fourier expansion of Apostol Frobenius–Euler polynomials and its applications. Adv. Differ. Equ. 2018, 67 (2018). https://doi.org/10.1186/s13662-018-1526-x3. Bayad, A.: Fourier expansion for Apostol Bernoulli, Apostol Euler and Apostol Genocchi polynomials. Math. Comput. 80, 2219–2221 (2011). https://doi.org/10.1090/S0025-5718-2011-02476-24. Bayad, A., Kim, T.: Identities for Apostol-type Frobenius–Euler polynomiasl resulting from the study of a nonlinear operator. Russ. J. Math. Phys. 23, 164–171 (2016). https://doi.org/10.1134/S10619208160200235. Cangul, I.N., Cevik, A.S., Simsek, Y.: Generalization of q-Apostol-type Eulerian numbers and polynomials, and their interpolation functions. Adv. Stud. Contemp. Math. 25(2), 211–220 (2015)6. Carlitz, L.: Eulerian numbers and polynomials. Math. Mag. 32, 247–260 (1959). https://doi.org/10.2307/30292257. Conway, J.B.: Functions of One Complex Variables. Springer, Berlin (1978)8. Cristina, B., Roberto, B.: Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli and Apostol–Euler polynomialsv. Adv. Differ. Equ. 2020, 346 (2020). https://doi.org/10.1186/s13662-020-02802-x9. Follan, G.: Fourier Analysis and Its Applications (1992)10. Kim, T.: An identity of the symmetry for the Frobenius–Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp. Rocky Mt. J. Math. 41, 239–247 (2011)11. Kucukoglu, I., Simsek, Y.: Identities and relations on the q-Apostol type Frobenius–Euler numbers and polynomials. J. Korean Math. Soc. 56(1), 265–284 (2019). https://doi.org/10.4134/JKMS.j18018512. Kucukoglu, I., Simsek, Y., Srivastava, H.M.: A new family of Lerch-type zeta functions interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials. Quaest. Math. 42 465–478 (2019). https://doi.org/10.2989/16073606.2018.145992513. Kurt, B., Simsek, Y.: On the generalized Apostol-type Frobenius–Euler polynomials. Adv. Differ. Equ. 2013, 1 (2013). https://doi.org/10.1186/1687-1847-2013-114. Luo, Q.: Fourier expansion and integral representations for the Apostol Bernoulli and Apostol Euler polynomials. Math. Comput. 78, 2193–2208 (2009)15. Luo, Q.-M.: Extensions of the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 48, 291–309 (2011)16. Quintana, Y., Ramírez, W., Urieles, A.: Euler matrices and their algebraic properties revisited. Appl. Math. Inf. Sci. 14(4), 583–596 (2020). https://doi.org/10.18576/amis/14040717. Ramírez, W., Ortega, M., Urieles, A.: New generalized Apostol Frobenius–Euler polynomials and their matrix approach. Kragujev. J. Math. 45(3), 393–407 (2021)18. Simsek, Y.: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. Fixed Point Theory Appl. 2013(87), 1 (2013). https://doi.org/10.1186/1687-1812-2013-8719. Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)20. Srivastava, H.M., Kurt, B., Simsek, Y.: Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms Spec. Funct. 23(12), 919–938 (2012). https://doi.org/10.1080/10652469.2011.64362721. Yilmaz, S.: Generating functions for q-Apostol type Frobenius–Euler numbers and polynomials. Axioms 1(3), 395–403 (2012). https://doi.org/10.3390/axioms1030395PublicationORIGINALFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdfFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdfapplication/pdf1509460https://repositorio.cuc.edu.co/bitstreams/18787330-7b33-4398-9d11-985d823e13f5/download7bcff39f2a974e7da3115aabdfd2d9dfMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.cuc.edu.co/bitstreams/3336be6f-94ed-49bc-ba14-02e5cad6aa3b/download42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83196https://repositorio.cuc.edu.co/bitstreams/9429157d-d872-4594-9956-f7d176bd7f36/downloade30e9215131d99561d40d6b0abbe9badMD53THUMBNAILFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.jpgFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.jpgimage/jpeg52920https://repositorio.cuc.edu.co/bitstreams/f0eb1ff1-fa37-470a-83ae-266026ddcd97/download4f22d98e515755a505c451b1a911d75aMD54THUMBNAILFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.jpgFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.jpgimage/jpeg52920https://repositorio.cuc.edu.co/bitstreams/617633e6-4e8d-4314-9988-1b68cf773dd6/download4f22d98e515755a505c451b1a911d75aMD54TEXTFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.txtFourier expansion and integral representation generalized Apostol-type Frobenius–Euler.pdf.txttext/plain30574https://repositorio.cuc.edu.co/bitstreams/d16090c9-130c-4685-a48c-e20555853e2f/download0cd99bc72be91e085b55497e4dae8e29MD5511323/7140oai:repositorio.cuc.edu.co:11323/71402024-09-17 10:16:19.703http://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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

Publicaciones similares