The essential purpose of this paper is to give some novel symmetric identities for generalized Carlitz's twisted $q$-Euler polynomials attached to $\chi$ based on the fermionic $p$-adic invariant integral on $Z_p$ under $S_4$.
[1] Apostol T. M., Introduction to Analytic Number Theory, New York, Splinger-Verlag, 1976.
[2] Araci S., Acikgoz M., Bagdasaryan A., Sen E., The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials, Turkish J. Anal. Number Theory, 1(2013), no. 1, 1-3.
[3] Araci S., Duran U., Acikgoz M., Symmetric identities involving q-Frobenius-Euler polynomials under Sym (5), Turkish J. Anal. Number Theory, 3(2015), no. 3, 90-93.
[4] Choi J., Anderson P. J., and Srivastava H. M., \Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler Polynomials of Order n and the Multiple Hurwitz Zeta Function," Appl. Math. Comput., 199(2008), 723{737.
[5] Choi J., Anderson P. J., and Srivastava H. M., \Carlitz's q-Bernoulli and q-Euler Numbers and Polynomials and a Class of q-Hurwitz Zeta Functions," Appl. Math. Comput., 215(2009), 1185-1208.
[6] Dolgy D. V., Jang Y. S., Kim T., Kwon H. I., Seo J.-J., Identities of symmetry for q-Euler polynomials derived from fermionic integral on Zp under symmetry group S3, Appl. Math. Sci., 8(2014), no. 113, 5599-5607.
[7] Dolgy D. V., Kim T., Rim S.-H., Lee S.-H, Some Symmetric Identities for h-Extension of q-Euler Polynomials
[8] Duran U., Acikgoz M., Esi A., Araci S., Some new symmetric identities involving q-Genocchi polynomials under S4, J. Math. Anal., 6(2015), no. 4, 22-31.
[9] Duran U., Acikgoz M., Araci S., Symmetric identities involving weighted q-Genocchi polyno mials under S4, Proc. Jangjeon Math. Soc., 18(2015), No. 4, 455-465.
[10] Duran U., Acikgoz M., New identities for Carlitz's twisted (h; q)-Euler polynomials under symmetric group of degree n, J. Anal. Number Theory, (2016), 4(2016), no. 2 133-137.
[11] Kim T., On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = 1, J. Math. Anal. Appl., 331(2007), 779-792.
[12] Kim T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp, Russ. J. Math. Phys., 16(2009), no. 4, 484-491.
[13] Kim T., A note on the p-adic invariant integral in the rings of p-adic integers, arXiv:math/0606097v1 [math.NT] (2006).
[14] Ozden H., Simsek Y., and Cangul I. N., Euler polynomials associated with p-adic q-Euler measure, Gen. Math., 15(2007), No. 2-3, 24-37.
[15] Rim S. H., Kim T., Lee S. H., Symmetric identities of generalized (h; q)-Euler polynomials under third dihedral group, Appl. Math. Sci., 8(2014), no. 145, 7207-7212.
[16] Ryoo C. S., Symmetric properties for Carlitz's twisted (h; q)-Euler polynomials associated with p-adic q-integral on Zp, Internat. J. Math. Anal., Vol. 9 (2015), no. 40, 1947 - 1953.
[17] Ryoo C. S., Symmetric properties for Carlitz's twisted q-Euler numbers and polynomials associated with p-adic integral on Zp, Internat. J. Math. Anal., 9 (2015), no. 83, 4129 - 4134.
[18] Seo J. J., Kim T., Identities of symmetry for generalized q{Euler polynomials attached to ksi under symmetric group S4, Adv. Stud. Theoret. Phys., 9(2015), no. 8, 353-359.
[19] Simsek Y., Complete sum of products of (h; q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl., 16(2010), no. 11, 1331-1348.
[20] Srivastava H. M., Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci., 5(2011), 390-444.
[21] Srivastava H. M., Kim T., and Simsek Y., q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math Phys., 12(2005), 241{268.
[1] Apostol T. M., Introduction to Analytic Number Theory, New York, Splinger-Verlag, 1976.
[2] Araci S., Acikgoz M., Bagdasaryan A., Sen E., The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials, Turkish J. Anal. Number Theory, 1(2013), no. 1, 1-3.
[3] Araci S., Duran U., Acikgoz M., Symmetric identities involving q-Frobenius-Euler polynomials under Sym (5), Turkish J. Anal. Number Theory, 3(2015), no. 3, 90-93.
[4] Choi J., Anderson P. J., and Srivastava H. M., \Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler Polynomials of Order n and the Multiple Hurwitz Zeta Function," Appl. Math. Comput., 199(2008), 723{737.
[5] Choi J., Anderson P. J., and Srivastava H. M., \Carlitz's q-Bernoulli and q-Euler Numbers and Polynomials and a Class of q-Hurwitz Zeta Functions," Appl. Math. Comput., 215(2009), 1185-1208.
[6] Dolgy D. V., Jang Y. S., Kim T., Kwon H. I., Seo J.-J., Identities of symmetry for q-Euler polynomials derived from fermionic integral on Zp under symmetry group S3, Appl. Math. Sci., 8(2014), no. 113, 5599-5607.
[7] Dolgy D. V., Kim T., Rim S.-H., Lee S.-H, Some Symmetric Identities for h-Extension of q-Euler Polynomials
[8] Duran U., Acikgoz M., Esi A., Araci S., Some new symmetric identities involving q-Genocchi polynomials under S4, J. Math. Anal., 6(2015), no. 4, 22-31.
[9] Duran U., Acikgoz M., Araci S., Symmetric identities involving weighted q-Genocchi polyno mials under S4, Proc. Jangjeon Math. Soc., 18(2015), No. 4, 455-465.
[10] Duran U., Acikgoz M., New identities for Carlitz's twisted (h; q)-Euler polynomials under symmetric group of degree n, J. Anal. Number Theory, (2016), 4(2016), no. 2 133-137.
[11] Kim T., On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = 1, J. Math. Anal. Appl., 331(2007), 779-792.
[12] Kim T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp, Russ. J. Math. Phys., 16(2009), no. 4, 484-491.
[13] Kim T., A note on the p-adic invariant integral in the rings of p-adic integers, arXiv:math/0606097v1 [math.NT] (2006).
[14] Ozden H., Simsek Y., and Cangul I. N., Euler polynomials associated with p-adic q-Euler measure, Gen. Math., 15(2007), No. 2-3, 24-37.
[15] Rim S. H., Kim T., Lee S. H., Symmetric identities of generalized (h; q)-Euler polynomials under third dihedral group, Appl. Math. Sci., 8(2014), no. 145, 7207-7212.
[16] Ryoo C. S., Symmetric properties for Carlitz's twisted (h; q)-Euler polynomials associated with p-adic q-integral on Zp, Internat. J. Math. Anal., Vol. 9 (2015), no. 40, 1947 - 1953.
[17] Ryoo C. S., Symmetric properties for Carlitz's twisted q-Euler numbers and polynomials associated with p-adic integral on Zp, Internat. J. Math. Anal., 9 (2015), no. 83, 4129 - 4134.
[18] Seo J. J., Kim T., Identities of symmetry for generalized q{Euler polynomials attached to ksi under symmetric group S4, Adv. Stud. Theoret. Phys., 9(2015), no. 8, 353-359.
[19] Simsek Y., Complete sum of products of (h; q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl., 16(2010), no. 11, 1331-1348.
[20] Srivastava H. M., Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci., 5(2011), 390-444.
[21] Srivastava H. M., Kim T., and Simsek Y., q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math Phys., 12(2005), 241{268.
Duran, U., & Acıkgoz, M. (2017). NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$. Konuralp Journal of Mathematics, 5(1), 49-55.
AMA
Duran U, Acıkgoz M. NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$. Konuralp J. Math. Nisan 2017;5(1):49-55.
Chicago
Duran, UGUR, ve MEHMET Acıkgoz. “NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$”. Konuralp Journal of Mathematics 5, sy. 1 (Nisan 2017): 49-55.
EndNote
Duran U, Acıkgoz M (01 Nisan 2017) NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$. Konuralp Journal of Mathematics 5 1 49–55.
IEEE
U. Duran ve M. Acıkgoz, “NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$”, Konuralp J. Math., c. 5, sy. 1, ss. 49–55, 2017.
ISNAD
Duran, UGUR - Acıkgoz, MEHMET. “NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$”. Konuralp Journal of Mathematics 5/1 (Nisan 2017), 49-55.
JAMA
Duran U, Acıkgoz M. NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$. Konuralp J. Math. 2017;5:49–55.
MLA
Duran, UGUR ve MEHMET Acıkgoz. “NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$”. Konuralp Journal of Mathematics, c. 5, sy. 1, 2017, ss. 49-55.
Vancouver
Duran U, Acıkgoz M. NOVEL IDENTITIES INVOLVING GENERALIZED CARLITZ’S TWISTED $q$-EULER POLYNOMIALS ATTACHED TO $\chi$ UNDER $S_4$. Konuralp J. Math. 2017;5(1):49-55.