Banu Yilmaz - Academia.edu (original) (raw)

Papers by Banu Yilmaz

Research paper thumbnail of Some properties of appell polynomials

This thesis consists of five chapters. The first Chapter gives general information about the thes... more This thesis consists of five chapters. The first Chapter gives general information about the thesis. In the second Chapter, some preliminaries and auxilary results that are used throughout the thesis are given. The original parts of the thesis are Chapters 3, 4 and 5 which are established from [35], [46] and [48]. In Chapter three, extended 2D Bernoulli and 2D Euler polynomials are introduced. Moreover, some recurrence relations are given. Differential, integrodifferential and partial differential equations of the extended 2D Bernoulli and the extended 2D Euler polynomials are obtained by using the factorization method. The special cases reduces to differential equation of the usual Bernoulli and Euler polynomials. Note that the results for the usual 2D Euler polynomials are new. In Chapter four, we consider Hermite-based Appell polynomials and give partial differential equations of them. In the special cases, we present the recurrence relation, differential, integro-differential and partial differential equations of the Hermite-based Bernoulli and Hermite-based Euler polynomials. In Chapter five, introducing k-times shift operators, factorization method is generalized. The differential equations of the Appell polynomials are obtained. For the special case k = 2, differential equation of Bernoulli and Hermite polynomials are exhibited.

Research paper thumbnail of Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Filomat, 2014

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell poly... more Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756-764] defined the Hermite-based Appell polynomials by G(x, y, z, t) := A(t)?exp(xt + yt2 + zt3) = ??,n=0 HAn(x, y, z) tn/n! and investigated their many interesting properties and characteristics by using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials via the factorization method. Furthermore, we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.

Research paper thumbnail of The extended Mittag-Leffler function and its properties

Journal of Inequalities and Applications, 2014

In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functi... more In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.

Research paper thumbnail of Differential equations for the extended 2D Bernoulli and Euler polynomials

Advances in Difference Equations, 2013

In this paper, we introduce the extended 2D Bernoulli polynomials by t α (e t-1) α c xt+yt j = ∞ ... more In this paper, we introduce the extended 2D Bernoulli polynomials by t α (e t-1) α c xt+yt j = ∞ n=0 B (α,j) n (x, y, c) t n n! and the extended 2D Euler polynomials by 2 α (e t + 1) α c xt+yt j = ∞ n=0 E (α,j) n (x, y, c) t n n! , where c > 1. By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2D Bernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415-428, 2004), in the special case c = e, α = 1. On the other hand, all the results for the second family are believed to be new, even in the case c = e, α = 1. Finally, we give some open problems related with the extensions of the above mentioned polynomials.

Research paper thumbnail of A set of finite order differential equations for the Appell polynomials

Journal of Computational and Applied Mathematics, 2014

Let {R n (x)} ∞ n=0 denote the set of Appell polynomials which includes, among others, Hermite, B... more Let {R n (x)} ∞ n=0 denote the set of Appell polynomials which includes, among others, Hermite, Bernoulli, Euler and Genocchi polynomials. In this paper, by introducing the generalized factorization method, for each k ∈ N, we determine the differential operator  L (x) n,k  ∞ n=0 such that L (x) n,k (R n (x)) = λ n,k R n (x), where λ n,k = (n+k)! n! − k!. The special case k = 1 reduces to the result obtained in [M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231-237]. The differential equations for the Hermite and Bernoulli polynomials are exhibited for the case k = 2.

Research paper thumbnail of Some properties of appell polynomials

This thesis consists of five chapters. The first Chapter gives general information about the thes... more This thesis consists of five chapters. The first Chapter gives general information about the thesis. In the second Chapter, some preliminaries and auxilary results that are used throughout the thesis are given. The original parts of the thesis are Chapters 3, 4 and 5 which are established from [35], [46] and [48]. In Chapter three, extended 2D Bernoulli and 2D Euler polynomials are introduced. Moreover, some recurrence relations are given. Differential, integrodifferential and partial differential equations of the extended 2D Bernoulli and the extended 2D Euler polynomials are obtained by using the factorization method. The special cases reduces to differential equation of the usual Bernoulli and Euler polynomials. Note that the results for the usual 2D Euler polynomials are new. In Chapter four, we consider Hermite-based Appell polynomials and give partial differential equations of them. In the special cases, we present the recurrence relation, differential, integro-differential and partial differential equations of the Hermite-based Bernoulli and Hermite-based Euler polynomials. In Chapter five, introducing k-times shift operators, factorization method is generalized. The differential equations of the Appell polynomials are obtained. For the special case k = 2, differential equation of Bernoulli and Hermite polynomials are exhibited.

Research paper thumbnail of Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Filomat, 2014

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell poly... more Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756-764] defined the Hermite-based Appell polynomials by G(x, y, z, t) := A(t)?exp(xt + yt2 + zt3) = ??,n=0 HAn(x, y, z) tn/n! and investigated their many interesting properties and characteristics by using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials via the factorization method. Furthermore, we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.

Research paper thumbnail of The extended Mittag-Leffler function and its properties

Journal of Inequalities and Applications, 2014

In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functi... more In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.

Research paper thumbnail of Differential equations for the extended 2D Bernoulli and Euler polynomials

Advances in Difference Equations, 2013

In this paper, we introduce the extended 2D Bernoulli polynomials by t α (e t-1) α c xt+yt j = ∞ ... more In this paper, we introduce the extended 2D Bernoulli polynomials by t α (e t-1) α c xt+yt j = ∞ n=0 B (α,j) n (x, y, c) t n n! and the extended 2D Euler polynomials by 2 α (e t + 1) α c xt+yt j = ∞ n=0 E (α,j) n (x, y, c) t n n! , where c > 1. By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2D Bernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415-428, 2004), in the special case c = e, α = 1. On the other hand, all the results for the second family are believed to be new, even in the case c = e, α = 1. Finally, we give some open problems related with the extensions of the above mentioned polynomials.

Research paper thumbnail of A set of finite order differential equations for the Appell polynomials

Journal of Computational and Applied Mathematics, 2014

Let {R n (x)} ∞ n=0 denote the set of Appell polynomials which includes, among others, Hermite, B... more Let {R n (x)} ∞ n=0 denote the set of Appell polynomials which includes, among others, Hermite, Bernoulli, Euler and Genocchi polynomials. In this paper, by introducing the generalized factorization method, for each k ∈ N, we determine the differential operator  L (x) n,k  ∞ n=0 such that L (x) n,k (R n (x)) = λ n,k R n (x), where λ n,k = (n+k)! n! − k!. The special case k = 1 reduces to the result obtained in [M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231-237]. The differential equations for the Hermite and Bernoulli polynomials are exhibited for the case k = 2.