[Paper Review] Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function

B. Ramakrishnan, Brundaban Sahu|ArXiv.org|Nov 22, 2007

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper presents a novel proof of van der Pol-type identities for Ramanujan's tau function using Rankin-Cohen brackets and quasimodular forms, offering new convolution identities involving divisor functions and deriving congruences. The method provides alternative derivations of known results and establishes previously unproven identities through modular and quasimodular form theory.

ABSTRACT

We use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier and D. Niebur on the van der Pol type identities for the Ramanujan's tau function. As consequences we obtain convolution sums and congruence relations involving the divisor functions.

Motivation & Objective

• To provide a new proof of van der Pol-type identities for the Ramanujan tau function using Rankin-Cohen brackets on modular and quasimodular forms.
• To derive new convolution identities for the tau function involving sums of products of divisor functions σ₃(n), σ₅(n), σ₇(n), and σ₁₁(n).
• To establish congruences among divisor functions by analyzing the Fourier coefficients of modular and quasimodular forms.
• To re-derive Niebur’s identity for τ(n) using quasimodular forms and provide new identities not previously proven.
• To unify and extend known results on τ(n) by systematically applying the theory of Rankin-Cohen brackets and quasimodular forms.

Proposed method

• Utilizes Rankin-Cohen brackets of modular forms E₄ and E₆ to generate new identities for τ(n) by comparing Fourier coefficients.
• Applies the theory of quasimodular forms to express Δ(z) in terms of E₂ and its derivatives, enabling derivation of Niebur-type identities.
• Employs differential operators and transformation laws of E₂ under SL₂(ℤ) to verify identities involving DᵏE₂ and their products.
• Uses decomposition of quasimodular forms into Eisenstein series and derivatives to equate Fourier expansions and derive convolution identities.
• Eliminates intermediate forms (e.g., Δ(z), D⁴E₄) between multiple Rankin-Cohen bracket expressions to derive new identities.
• Applies known dimension formulas and modularity properties (e.g., dim S₁₆(1) = 1) to equate modular forms and extract coefficient identities.

Experimental results

Research questions

• RQ1 Can Rankin-Cohen brackets be used to re-derive known van der Pol-type identities for the Ramanujan tau function?
• RQ2 What new convolution identities for τ(n) can be generated using Rankin-Cohen brackets involving σ₃, σ₅, σ₇, and σ₁₁ functions?
• RQ3 Can Niebur’s identity for τ(n) be re-proven using the theory of quasimodular forms and E₂ derivatives?
• RQ4 What new congruences among divisor functions arise from the derived identities involving τ(n) and convolution sums?
• RQ5 How do the identities derived from quasimodular forms compare in structure and novelty to those from modular form bracket constructions?

Key findings

• The paper establishes four new identities for τ(n) involving σ₇(n), σ₃(n), and convolution sums of σ₃(m)σ₃(n−m), including Theorem 2.1(i)–(iv).
• It derives five new identities for τ(n) involving σ₉(n), σ₅(n), σ₃(n), and convolution sums of σ₃(m)σ₅(n−m), as in Theorem 2.2(i)–(v).
• A new identity is proven for τ(n) in terms of σ₁₁(n), σ₅(n), and a convolution sum of σ₅(m)σ₅(n−m), as stated in Theorem 2.3.
• The paper provides two new identities involving σ₁₁(n), σ₃(n), σ₇(n), and convolution sums of σ₃(m)σ₇(n−m), as in Theorem 2.4(i)–(ii).
• A new identity is derived for τ(n) using only σ(n) and convolution sums of σ(m)σ(n−m), as in Theorem 2.5(i)–(iv), extending Niebur’s original result.
• The paper proves identities (25), (30), and (31) relating convolution sums of σ(m)σ(n−m) to σ(n) and σ₃(n), which are used to derive further identities in Theorem 2.9.