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

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

Keywords

• Frobenius-Euler polynomials,Frobenius-Genocchi polynomials,differential equations,recurrence relations

References

1. 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.
2. 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.
3. Bretti,G, Ricci, P.E: Euler Polynomials and the Related Quadrature Rule, Georgian Mathematical Journal, 8 (2001), No:3, 447-453.
4. Bretti, G, Ricci, P.E: Multidimensional extension of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics 8 (3) (2004), 415–428.
5. 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.
6. Cheikh, Y.B: Some Results on quasi-monomiality, Applied Mathematics and Computation 141 (2003) 63-76.
7. 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.
8. 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).

9. 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
10. Dattoli, G, Torre, A, Mazzacurati,G: Quasi-monomials and isospectral problems, Nuovo Cimento B 112 (1997) 133-138.
11. 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.
12. 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.
13. Infeld, L, Hull,T.E: The factorization method, Rev.Mod.Phys., 23(1951), 21-68.
14. 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.
15. Kim, D,S, Kim, T: Some new identities of Frobenius-Euler numbers and polynomials, Journal of Inequalities and Applications 307(2012),10 pp.
16. Kim, D,S, Kim, T: Some identities of Frobenius-Euler polynomials arising from umbral calculus, Advances in Difference Equations 1 (2012), 196.
17. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, Journal of Number Theory 132 (2012), No:12, 2854-2865.
18. Kurt, B, Simsek, Y: On the generalized Apostol-type Frobenius-Euler polynomials, Advances in Difference Equations, 2013,1.
19. 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.
20. ¨Ozarslan, M. A: Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Computers and Mathematics with Applications, 62(2011), 2452-2462.
21. ¨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.
22. ¨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.
23. Pinter, A, Srivastava, H.M: Addition Theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math. 85(2013), 483-495.
24. Sheffer, I.M: A Differential Equation for Appell Polynomial, American Mathematical Society, 1935.
25. 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.
26. Smirnov, Y, Turbiner, A : Hidden SL2-algebra of finite difference equations, Mod. Phys. Lett. A 10 (1995) 1795-1801.
27. 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.
28. 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.
29. 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.
30. Yılmaz Yas¸ar, B, ¨Ozarslan, M.A: Differential equations for the extended 2D Bernoulli and Euler polynomials, Advances in Difference Equations 107(2013).