New Proofs of Some Modular Equations in the Theory of Signature 3

This paper deals with the construction of explicit formulas for sum of consecutive balancing numbers, consecutive even/odd balancing numbers, squares of consecutive balancing numbers, squares of consecutive even/odd balancing numbers and pronic product of balancing numbers. Sums of these numbers with alternative signs give beautiful results. When indices are in arithmetic progression, ratios of sums and differences follow certain interesting patterns.