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Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a Pi_{q}-identity of Gosper.
Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $lvert xrvert^s$ and $log lvert xrvert$ are multiples of $lvert xrvert^{s-k}$ and $lvert xrvert^{-k}$, respectively. In the present
We give unified modular proofs to all of Gospers identities on the $q$-constant $Pi_q$. We also confirm Gospers observation that for any distinct positive integers $n_1,cdots,n_m$ with $mgeq 3$, $Pi_{q^{n_1}}$, $cdots$, $Pi_{q^{n_m}}$ satisfy a nonze
In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a va
We prove the height two case of a conjecture of Hovey and Strickland that provides a $K(n)$-local analogue of the Hopkins--Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by
We prove that $$max_{t in [-pi,pi]}{|Q(t)|} leq T_{2n}(sec(s/4)) = frac 12 ((sec(s/4) + tan(s/4))^{2n} + (sec(s/4) - tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m({t in [-