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In the first series "Special functions and three term recurrence formula(3TRF)", I show how to obtain power series solutions of Heun, Grand ConfluentHypergeoemtric (GCH), Mathieu and Lame equations for an infinite series and apolynomial of type 1. The method of proof for an infinite series and apolynomial of type 1 in the 3-term recurrence relation is called as three termrecurrence formula (3TRF). And integral forms and generating functions of theabove 4 equations are constructed analytically.  In the second series "Special functions and reversible three-term recurrenceformula (R3TRF)", I show how to obtain (1) power series solutions, (2) integralsolutions and (3) generating functions of 5 equations (Heun, GCH, Mathieu, Lameand Confluent Heun (CH) equations) for an infinite series and a polynomial oftype 2. The method of proof for an infinite series and a polynomial of type 2in the 3-term recurrence relation is called as reversible three term recurrenceformula (R3TRF).  In this series I show how to obtain the mathematical formula for a polynomialof type 3, designated as "complete polynomial." The complete polynomial has twodifferent types which are (1) the first species complete polynomial and (2) thesecond species complete polynomial. The former is applicable if there are onlyone eigenvalue in B_n term and an eigenvalue in A_n term. And the latter isapplicable if there are two eigenvalues in B_n term and an eigenvalue in A_nterm. By applying 3TRF and R3TRF, I generalize the 3-term recurrence relationin 5 equations (Heun, GCH, Lame, CH and Double Confluent Heun equations) forcomplete polynomials of two types in the form of a power series expansion.

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