On Incomplete Symmetric Orthogonal Polynomials of Jacobi Type

In this paper, by using the extended Sturm-Liouville theorem for symmetric functions, we introduce the following differential equation ( ) 0 ) ( Φ ) 2 ) 1 ( 1 ( ) ( Φ 1 ) 1 ( 2 ) ( Φ ) 1 ( 2 2 2 2 = − − + + + ′ − + − + + − ′ ′ − x x x m a x mb a x x x x n n m n n m n m γ β α , in which ) 1 2 2 2 ( 2 + − + − = m a s s β ; ) 1 )( 1 2 ( 2 ) 1 2 2 2 ( 2 + − + + − + − + = m a r r m a s s γ and ( )( ) ) ) 1 ( 1 )( 2 / ) 1 ( ( 2 1 2 2 ) ) 1 ( 1 )( 2 / ) 1 ( ( 2 n n n m s r mb a s mn m s r s mn − − − + − + + + + + − − − + − + + = α and show that one of its basic solutions is a class of incomplete symmetric polynomials orthogonal with respect to the weight function b m a x x ) 1 ( 2 2 − on [-1,1]. We also obtain the norm square value of this orthogonal class.