Determine coefficients in the conjectured arcsin/arctan representation of 2F1(2,2; 9/(4k); 1/(4k))

Determine explicit closed-form formulas for the coefficient sequences a_k, b_k, and c_k, as functions of the natural number k, in the conjectured identity expressing the Gauss hypergeometric function 2F1(2,2; 9/(4k); 1/(4k)) as a linear combination a_k + b_k·arcsin(1/√k) + c_k·arctan(√(4k − 1)).

Background

In Section 4, the authors evaluate several instances of the Gauss hypergeometric function 2F1 with parameters (2,2; 9/(4k); 1/(4k)) for specific values k = 1, 2, 3. These evaluations yield expressions involving combinations of constants together with arcsine and arctangent terms whose arguments depend on k (notably arcsin(1/√k) and arctan(√(4k − 1)).

Observing a recurring structure across these cases, the authors propose a general pattern and formulate a conjecture for all natural k. While the functional form is hypothesized, they explicitly note that determining the general coefficient expressions a_k, b_k, and c_k remains unresolved.