What is the derivative of Rogers-Ramanujan Continued Fraction?
In the answer of upper link, Nikos Bagis claimed that $$ R'(q)=5^{-1}q^{-5/6}f(-q)^4R(q)\sqrt[6]{R(q)^{-5}-11-R(q)^5}\textrm{, }:(d1)$$
Where can I get a proof for this formula?
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A proof is given in Berndt, Ramanujan's Notebooks, Part III, page 267, equation (11.6). See also Andrews and Berndt, Ramanujan's Lost Notebook, Part I, page 334, – Somos Jul 25 '18 at 18:15
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The radical on right is same as $q^{-1/6}f(-q)/f(-q^5)$ so that the derivative is equal to $f^{5}(-q)R(q)/5qf(-q^5)$ and thus it is sufficient to prove that $(\log R(q)) '=f^{5}(-q)/5qf(-q^5)$. – Paramanand Singh Jul 26 '18 at 08:59
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A proof for the identity in my previous comment is now available at https://math.stackexchange.com/a/5046120/72031. – Paramanand Singh Mar 16 '25 at 02:47