Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete polynomials using 3-term and reversible 3-term recurrence
  formulas (3TRF and R3TRF)   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.

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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat