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Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations

Year 2015, Volume: 3 Issue: 2, 172 - 180, 19.01.2015

Abstract

In the present paper, we obtain differential equations of Frobenius-Euler polynomials by using quasi-monomiality principle.Furthermore, we introduce Frobenius-Genocchi polynomials and obtain some recurrence relation and some differential equations

References

  • Aracı, S, Ac¸ıkg¨oz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Advanced Studies in Contemporary Mathematics 22 (2012), No:3, 399-406.
  • Aracı, S, Ac¸ıkg¨oz, M, Park, K.H, Jolany, H: On the Unification of Two Families of Multiple Twisted Type Polynomials by using p-Adic q-integral at q =−1, Bulletin of the Malaysian Mathematical Sciences Society, 37(2) (2014), 543-554.
  • Bretti,G, Ricci, P.E: Euler Polynomials and the Related Quadrature Rule, Georgian Mathematical Journal, 8 (2001), No:3, 447-453.
  • Bretti, G, Ricci, P.E: Multidimensional extension of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics 8 (3) (2004), 415–428.
  • Cesarano, C: Monomiality principle and Legendre polynomails, in: G. Dattoli, H. M. Srivastava, C. Cesarano(Eds.), Advanced Special Functions and Integration Methods(Proceeding of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 18-23 June 2000). Aracne Editrice, Rome, 2001, pp. 83-95.
  • Cheikh, Y.B: Some Results on quasi-monomiality, Applied Mathematics and Computation 141 (2003) 63-76.
  • Choi, J, Srivastava, H.M: Series involving the zeta functions and a family of generalized Goldbach-Euler series, Amer. Math. Monthly 121 (2014), 229-236.
  • Choi, J, Kim, D.S, Kim, T, Kim, Y.H: A note on some identities of Frobenius-Euler numbers and polynomials, International Journal of Mathematics and Mathematical Sciences, (2012).
  • Dattoli, G: Hermite-Bessel, Laguerre-Bessel functions: a by-product of the monomiality principle, in: D. Cocolicchio, G. Dattoli, H.M. Srivastava (Eds.), Advanced Topics in Mathematics and Physics; Melfi, 9-12 May 1999), Aracne Editrice, Rome, 2000, pp. 147-1
  • Dattoli, G, Torre, A, Mazzacurati,G: Quasi-monomials and isospectral problems, Nuovo Cimento B 112 (1997) 133-138.
  • He, M.X , Ricci, P.E: Differential equation of Appell polynomials via the factorization method, Journal of Computational and Applied Mathematics 139(2), (2002), 231-237.
  • He, M.X, Ricci, P.E: Differential equations of some classes of special functions via the factorization method, Journal of Computational Analysis and Applications 6(2004), No:3.
  • Infeld, L, Hull,T.E: The factorization method, Rev.Mod.Phys., 23(1951), 21-68.
  • Khan, S, Yasmin, G, Khan, R, Hassan, N.A.M: Hermite-Based Appell Polynomials, Properties and Applications, Journal of Mathematical Analysis and Applications, 351(2009) 756-764.
  • Kim, D,S, Kim, T: Some new identities of Frobenius-Euler numbers and polynomials, Journal of Inequalities and Applications 307(2012),10 pp.
  • Kim, D,S, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus, Advances in Difference Equations 1 (2012), 196.
  • Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, Journal of Number Theory 132 (2012), No:12, 2854-2865.
  • Kurt, B, Simsek, Y: On the generalized Apostol-type Frobenius-Euler polynomials, Advances in Difference Equations, 2013,1.
  • Liu, H, Wang, W: Some Identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums, Discrete Mathematics, 309 (2009) 3346-3363.
  • ¨Ozarslan, M. A: Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Computers and Mathematics with Applications, 62(2011), 2452-2462.
  • ¨Ozarslan, M. A, Yılmaz Yas¸ar, B: A set of finite order differential equations for the Appell polynomials, Journal of Computational and Applied Mathematics, 259(2014) 108-116.
  • ¨Ozden, H, S¸ims¸ek, Y, Srivastava, H. M: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Computers and Mathematics with Applications 60(2010), 2779-2787.
  • Pinter, A, Srivastava, H.M: Addition Theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math. 85(2013), 483-495.
  • Sheffer, I.M: A Differential Equation for Appell Polynomial, American Mathematical Society, 1935.
  • Srivastava, H.M, ¨Ozarslan, M.A, Yılmaz Yas¸ar, B: Some Families of Differential Equations Associated with the Hermite-Based Appell Polynomials and Other Classes of Hermite-Based Polynomials, Filomat, 4(2014), No:28, 695-708.
  • Smirnov, Y, Turbiner, A : Hidden SL2-algebra of finite difference equations, Mod. Phys. Lett. A 10 (1995) 1795-1801.
  • Srivastava,H.M, ¨Ozarslan, M.A and Kaano˘glu, C: Some families of generating functions for a certain class of three-variable polynomials, Integral Transforms and Special Functions, 21(2012) No:12, 885-896.
  • S¸ims¸ek, Y, Bayad, A, Lokesha, V: q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions and q-Stirling numbers, Math. Methods Appl. Sci. 35 (2012), No: 8, 877-884.
  • S¸ims¸ek, Y, Kim, T, Park, D.W, Ro, Y.S, Jang, L.C, Rim, S.H : An explicit formula for the multiple Frobenius-Euler numbers and polynomials, Journal of Algebra, Number Theory and Applications 4 (2004), No: 3, 519-529.
  • Yılmaz Yas¸ar, B, ¨Ozarslan, M.A: Differential equations for the extended 2D Bernoulli and Euler polynomials, Advances in Difference Equations 107(2013).
Year 2015, Volume: 3 Issue: 2, 172 - 180, 19.01.2015

Abstract

References

  • Aracı, S, Ac¸ıkg¨oz, M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Advanced Studies in Contemporary Mathematics 22 (2012), No:3, 399-406.
  • Aracı, S, Ac¸ıkg¨oz, M, Park, K.H, Jolany, H: On the Unification of Two Families of Multiple Twisted Type Polynomials by using p-Adic q-integral at q =−1, Bulletin of the Malaysian Mathematical Sciences Society, 37(2) (2014), 543-554.
  • Bretti,G, Ricci, P.E: Euler Polynomials and the Related Quadrature Rule, Georgian Mathematical Journal, 8 (2001), No:3, 447-453.
  • Bretti, G, Ricci, P.E: Multidimensional extension of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics 8 (3) (2004), 415–428.
  • Cesarano, C: Monomiality principle and Legendre polynomails, in: G. Dattoli, H. M. Srivastava, C. Cesarano(Eds.), Advanced Special Functions and Integration Methods(Proceeding of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 18-23 June 2000). Aracne Editrice, Rome, 2001, pp. 83-95.
  • Cheikh, Y.B: Some Results on quasi-monomiality, Applied Mathematics and Computation 141 (2003) 63-76.
  • Choi, J, Srivastava, H.M: Series involving the zeta functions and a family of generalized Goldbach-Euler series, Amer. Math. Monthly 121 (2014), 229-236.
  • Choi, J, Kim, D.S, Kim, T, Kim, Y.H: A note on some identities of Frobenius-Euler numbers and polynomials, International Journal of Mathematics and Mathematical Sciences, (2012).
  • Dattoli, G: Hermite-Bessel, Laguerre-Bessel functions: a by-product of the monomiality principle, in: D. Cocolicchio, G. Dattoli, H.M. Srivastava (Eds.), Advanced Topics in Mathematics and Physics; Melfi, 9-12 May 1999), Aracne Editrice, Rome, 2000, pp. 147-1
  • Dattoli, G, Torre, A, Mazzacurati,G: Quasi-monomials and isospectral problems, Nuovo Cimento B 112 (1997) 133-138.
  • He, M.X , Ricci, P.E: Differential equation of Appell polynomials via the factorization method, Journal of Computational and Applied Mathematics 139(2), (2002), 231-237.
  • He, M.X, Ricci, P.E: Differential equations of some classes of special functions via the factorization method, Journal of Computational Analysis and Applications 6(2004), No:3.
  • Infeld, L, Hull,T.E: The factorization method, Rev.Mod.Phys., 23(1951), 21-68.
  • Khan, S, Yasmin, G, Khan, R, Hassan, N.A.M: Hermite-Based Appell Polynomials, Properties and Applications, Journal of Mathematical Analysis and Applications, 351(2009) 756-764.
  • Kim, D,S, Kim, T: Some new identities of Frobenius-Euler numbers and polynomials, Journal of Inequalities and Applications 307(2012),10 pp.
  • Kim, D,S, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus, Advances in Difference Equations 1 (2012), 196.
  • Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, Journal of Number Theory 132 (2012), No:12, 2854-2865.
  • Kurt, B, Simsek, Y: On the generalized Apostol-type Frobenius-Euler polynomials, Advances in Difference Equations, 2013,1.
  • Liu, H, Wang, W: Some Identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums, Discrete Mathematics, 309 (2009) 3346-3363.
  • ¨Ozarslan, M. A: Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Computers and Mathematics with Applications, 62(2011), 2452-2462.
  • ¨Ozarslan, M. A, Yılmaz Yas¸ar, B: A set of finite order differential equations for the Appell polynomials, Journal of Computational and Applied Mathematics, 259(2014) 108-116.
  • ¨Ozden, H, S¸ims¸ek, Y, Srivastava, H. M: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Computers and Mathematics with Applications 60(2010), 2779-2787.
  • Pinter, A, Srivastava, H.M: Addition Theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math. 85(2013), 483-495.
  • Sheffer, I.M: A Differential Equation for Appell Polynomial, American Mathematical Society, 1935.
  • Srivastava, H.M, ¨Ozarslan, M.A, Yılmaz Yas¸ar, B: Some Families of Differential Equations Associated with the Hermite-Based Appell Polynomials and Other Classes of Hermite-Based Polynomials, Filomat, 4(2014), No:28, 695-708.
  • Smirnov, Y, Turbiner, A : Hidden SL2-algebra of finite difference equations, Mod. Phys. Lett. A 10 (1995) 1795-1801.
  • Srivastava,H.M, ¨Ozarslan, M.A and Kaano˘glu, C: Some families of generating functions for a certain class of three-variable polynomials, Integral Transforms and Special Functions, 21(2012) No:12, 885-896.
  • S¸ims¸ek, Y, Bayad, A, Lokesha, V: q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions and q-Stirling numbers, Math. Methods Appl. Sci. 35 (2012), No: 8, 877-884.
  • S¸ims¸ek, Y, Kim, T, Park, D.W, Ro, Y.S, Jang, L.C, Rim, S.H : An explicit formula for the multiple Frobenius-Euler numbers and polynomials, Journal of Algebra, Number Theory and Applications 4 (2004), No: 3, 519-529.
  • Yılmaz Yas¸ar, B, ¨Ozarslan, M.A: Differential equations for the extended 2D Bernoulli and Euler polynomials, Advances in Difference Equations 107(2013).
There are 30 citations in total.

Details

Journal Section Articles
Authors

Banu Yılmaz Yaşar This is me

Mehmet Ali Özarslan This is me

Publication Date January 19, 2015
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Yaşar, B. . Y., & Özarslan, M. . A. (2015). Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations. New Trends in Mathematical Sciences, 3(2), 172-180.
AMA Yaşar BY, Özarslan MA. Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations. New Trends in Mathematical Sciences. January 2015;3(2):172-180.
Chicago Yaşar, Banu Yılmaz, and Mehmet Ali Özarslan. “Frobenius-Euler and Frobenius-Genocchi Polynomials and Their Differential Equations”. New Trends in Mathematical Sciences 3, no. 2 (January 2015): 172-80.
EndNote Yaşar BY, Özarslan MA (January 1, 2015) Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations. New Trends in Mathematical Sciences 3 2 172–180.
IEEE B. . Y. Yaşar and M. . A. Özarslan, “Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations”, New Trends in Mathematical Sciences, vol. 3, no. 2, pp. 172–180, 2015.
ISNAD Yaşar, Banu Yılmaz - Özarslan, Mehmet Ali. “Frobenius-Euler and Frobenius-Genocchi Polynomials and Their Differential Equations”. New Trends in Mathematical Sciences 3/2 (January 2015), 172-180.
JAMA Yaşar BY, Özarslan MA. Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations. New Trends in Mathematical Sciences. 2015;3:172–180.
MLA Yaşar, Banu Yılmaz and Mehmet Ali Özarslan. “Frobenius-Euler and Frobenius-Genocchi Polynomials and Their Differential Equations”. New Trends in Mathematical Sciences, vol. 3, no. 2, 2015, pp. 172-80.
Vancouver Yaşar BY, Özarslan MA. Frobenius-Euler and Frobenius-Genocchi Polynomials and their differential equations. New Trends in Mathematical Sciences. 2015;3(2):172-80.