Linear independence of certain numbers
Sign Up to like & getrecommendations! Published in 2018 at "Archiv der Mathematik"
DOI: 10.1007/s00013-018-1271-0
Abstract: Let $$k\ge 2$$k≥2, $$b\ge 2$$b≥2 and $$1\le a_1 read more here.
Keywords: sum infty; certain numbers; independence certain; linear independence ... See more keywords
Proofs of some conjectures of Chan and Wang on congruences for $$(q;q)_\infty ^{\frac{a}{b}}$$
Sign Up to like & getrecommendations! Published in 2020 at "Ramanujan Journal"
DOI: 10.1007/s11139-019-00207-3
Abstract: Recently, Chan and Wang proved numerous congruences satisfied by $$p_{a/b}(n)$$, where $$p_{a/b}(n)$$ denotes the coefficient of $$q^n$$ in the series expansion of $$(q;q)_\infty ^{\frac{a}{b}}$$. Moreover, they presented many conjectures on the congruences for $$p_{a/b}(n)$$. In… read more here.
Keywords: proofs conjectures; wang congruences; chan wang; conjectures chan ... See more keywords
Two $q$-analogues of Euler’s formula $\zeta (2)=\pi ^2/6$
Sign Up to like & getrecommendations! Published in 2019 at "Colloquium Mathematicum"
DOI: 10.4064/cm7686-11-2018
Abstract: It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\sum_{k=0}^\infty\frac{q^{2k-\lfloor(-1)^kk/2\rfloor}}{(1-q^{2k+1})^2} =\prod_{n=1}^\infty\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q| read more here.
Keywords: formula zeta; euler formula; two analogues; infty frac ... See more keywords