IJPAM: Volume 50, No. 1 (2009)

COMBINATIONS OF ORTHOGONAL POLYNOMIALS

Chrysi G. Kokologiannaki
Department of Mathematics
Faculty of Science
University of Patras
Patras, 26500, GREECE
e-mail: [email protected]


Abstract.Let $\{P_{n}(x)\}_{n=0}^{\infty}$ be an orthogonal sequence of polynomials. We consider two families of polynomials defined by $ Q_{n}(x)=AP_{n}(x)+BP_{n-1}(x)+CP_{n-2}(x)$, where $A,B,C$ are real numbers with $A\ne 0$ and $S_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+\Gamma_{n}P_{n}(x)$, where $A_{n}\ne 0, B_{n}$ and $\Gamma_{n}$ are real sequences. We find conditions that the polynomials $Q_{n}(x)$ and $S_{n}(x)$ are orthogonal, we derive the three-term recurrence relation which is satisfied by each of them and we illustrate the results by some examples. We also give some results concerning the zeros of the polynomials $Q_{n}(x)$ and $S_{n}(x)$.

Received: November 24, 2008

AMS Subject Classification: 33C45, 05E35

Key Words and Phrases: orthogonal polynomials, combination, zeros

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 1