On the Proofs of the Rogers-Ramanujan Identities

The celebrated Rogers-Ramanujan identities are familiar in two forms [52; pp. 33-48]. First as series-product identities: 1.1$$1 + \sum\limits_{n = 1}^\infty {\frac{{{q^{{n^2}}}}}{{\left( {1 - q} \right)\left( {1 - {q^2}} \right) \ldots \left( {1 - {q^n}} \right)}} = \prod\limits_{n = 0}^\infty {\frac{1}{{\left( {1 - {q^{5n + 1}}} \right)\left( {1 - {q^{5n + 4}}} \right)}}} } $$1.2$$1 + \sum\limits_{n = 1}^\infty {\frac{{{q^{{n^2} + n}}}}{{\left( {1 - q} \right)\left( {1 - {q^2}} \right) \ldots \left( {1 - {q^n}} \right)}} = \prod\limits_{n = 0}^\infty {\frac{1}{{\left( {1 - {q^{5n + 2}}} \right)\left( {1 - {q^{5n + 3}}} \right)}}} } $$