Matching coefficients in the series expansions of certain $q$-products and their reciprocals
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Publication:6381635
DOI10.1007/S11139-021-00534-4arXiv2110.15546MaRDI QIDQ6381635
Hirakjyoti Das, Nayandeep Deka Baruah
Publication date: 29 October 2021
Abstract: We show that the series expansions of certain -products have extit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let denote the Rogers-Ramanujan continued fraction having the well-known -product repesentation R(q)=dfrac{(q;q^5)_infty(q^4;q^5)_infty}{(q^2;q^5)_infty(q^3;q^5)_infty}. If �egin{align*} sum_{n=0}^{infty}alpha(n)q^n=dfrac{1}{R^5left(q
ight)}=left(sum_{n=0}^{infty}alpha^{prime}(n)q^n
ight)^{-1},\ sum_{n=0}^{infty}�eta(n)q^n=dfrac{R(q)}{Rleft(q^{16}
ight)}=left(sum_{n=0}^{infty}�eta^{prime}(n)q^n
ight)^{-1}, end{align*} then �egin{align*} alpha(5n+r)&=-alpha^{prime}(5n+r-2) quad rin{3,4},\ �eta(10n+r)&=-�eta^{prime}(10n+r-6) quad rin{7,9}. end{align*}
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