Exact Linear Representations of the Polynomial n(k) via Modular Forms: n(k) = A·r₂(m) + B·τ(m) + C·r₄(m)

Key Points

• This research aims to establish exact linear representations of the polynomial n(k) through the lens of modular forms.
• Utilized classical modular forms to derive linear representations of n(k) for specific integer values of k.
• Proved relationships for nine values of k, with integer coefficients connected through a universal constant A.
• Included explicit formulas for specific values of k and explored connections with divisor sum functions for integers.
• Discovered a universal constant A = -15 linking seven of the nine representations of n(k).
• Established exact formulas for n(k) for k values of -3, -1, 1, 2, 3, 5, 6, 7, 8.
• Demonstrated connections between n(k) and divisor sum functions σₐ(k) for k = 1, 2, 3, 4, 5.