Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)

Abstract: In the first series "Special functions and three term recurrence formula (3TRF)", I show how to obtain power series solutions of Heun, Grand Confluent Hypergeoemtric (GCH), Mathieu and Lame equations for an infinite series and a polynomial of type 1. The method of proof for an infinite series and a polynomial of type 1 in the 3-term recurrence relation is called as three term recurrence formula (3TRF). And integral forms and generating functions of the above 4 equations are constructed analytically. In the second series "Special functions and reversible three-term recurrence formula (R3TRF)", I show how to obtain (1) power series solutions, (2) integral solutions and (3) generating functions of 5 equations (Heun, GCH, Mathieu, Lame and Confluent Heun (CH) equations) for an infinite series and a polynomial of type 2. The method of proof for an infinite series and a polynomial of type 2 in the 3-term recurrence relation is called as reversible three term recurrence formula (R3TRF). In this series I show how to obtain the mathematical formula for a polynomial of type 3, designated as "complete polynomial." The complete polynomial has two different types which are (1) the first species complete polynomial and (2) the second species complete polynomial. The former is applicable if there are only one eigenvalue in B_n term and an eigenvalue in A_n term. And the latter is applicable if there are two eigenvalues in B_n term and an eigenvalue in A_n term. By applying 3TRF and R3TRF, I generalize the 3-term recurrence relation in 5 equations (Heun, GCH, Lame, CH and Double Confluent Heun equations) for complete polynomials of two types in the form of a power series expansion.