1. ### Generalized Chebyshev Polynomials Via Matrices and Combinatorial Forms of Their Derivatives

Inci GULTEKIN and Betul SAKIROGLU

1. Departmentof Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey.

2. Departmentof Mathematics, Science Institute, Atatürk University, 25240 Erzurum, Turkey.

Abstract: In this paper, we obtain the generalized Chebyshev polynomials Vn,m(x) and Ωn,m(x) via matrices and then we define new recurrence relation for the derivative of these polynomials. Also, we give combinatorial forms for the derivatives of these polynomials. Then we create tables for derivative polynomial with the help of combinatorial forms. Finally, we give examples showing how to write derivative polynomials easily.

Keywords: Chebyshev polynomials, Vieta-Fibonacci polynomials, Vieta-Lucas polynomials, determinant, matrix.

2. ### Stability Analysis of a Three Species Food Chain Model with Harvesting

Kondala Rao. K and Lakshmi Narayan. K

1. Research Scholar, Rayalaseema university, Kurnool.

2. Vignan Institute of Technology & Science, Deshmukhi (Vill), Hyderabad-508 284.

Abstract: The Present investigation is an analytical study of a three species syn-ecological model which comprises three species. Here first species (N1) ammensal on the second (N2), second species ammensal on the third (N3). In this model first species (N1) and third species (N3) are neutral to each other. And first (N1) and second species (N2) are harvested at a rate proportional to their population sizes.  All possible equilibrium points are identified and the stability of Interior equilibrium point is discussed by using Routh-Hurwitz criteria and the solutions are carried out. Further the global stability of the system is discussed by constructing a suitable Lyapunov function. The analytical results are supported by numerical simulation using Mat Lab.

Keywords: Ammensal, Neutral, Equilibrium Points, Lyapunov’s function and Routh-Hurwitz criteria.