On Incomplete Symmetric Orthogonal Polynomials of Laguerre Type

By using the extended Sturm-Liouville theorem for symmetric functions, we introduce the following differential equation ( ) 0 ) ( ) 2 ) 1 ( 1 ( ) ( 1 2 ) ( 2 2 2 = Φ − − + + + Φ′ − + − − Φ ′ ′ x x x m a x m x x x n n m n n m n 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 2 n n m s r s mn m − − − + − + + = α and show that one of its basis solutions is a class of incomplete symmetric polynomials orthogonal with respect to the weight function ) exp( 2 2 m a x x − on ) , ( ∞ −∞ . We also obtain the norm square value of this orthogonal class.