The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type.

The Heun–Askey–Wilson algebra is introduced through generators { X , W } and relations. These relations can be understood as an extension of the usual Askey–Wilson ones. A central element is given, and a canonical form of the Heun–Askey–Wilson algebra is presented. A homomorphism from the Heun–Askey–Wilson algebra to the Askey–Wilson one is identified. On the vector space of the polynomials in the variable x = z + z - 1 , the Heun operator of Askey–Wilson type realizing W can be characterized as the most general second-order q-difference operator in the variable z that maps polynomials of degree n in x = z + z - 1 into polynomials of degree n + 1 . [ABSTRACT FROM AUTHOR]