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 paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.
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 nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gospers identities. Meanwhile, several results on $Pi_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gospers identities using hauptmoduls for genus zero congruence subgroups.
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 vanishing identity which is stronger than their conjectures. Moreover we will also prove their conjectures in low genus cases.
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 Chai. We then use the Gross--Hopkins period map to verify Chais Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava $E$-theory is coherent, and that every finitely generated Morava module can be realized by a $K(n)$-local spectrum as long as $2p-2>n^2+n$. Finally, we deduce consequences of our results for descent of Balmer spectra.
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 $$m({t in [-pi,pi]: |Q(t)| leq 1}) geq 2pi-s,, qquad s in (0,2pi),,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A subset {Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(cos t) = cos(2nt)$ for $t in {Bbb R}$. This inequality is sharp. We also prove that $$max_{t in [-pi,pi]}{|Q(t)|} leq T_{2n}(sec(s/2)) = frac 12 ((sec(s/2) + tan(s/2))^{2n} + (sec(s/2) - tan(s/2))^{2n})$$ for every trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m({t in [-pi,pi]: |Q(t)| leq 1}) geq 2pi-s,, qquad s in (0,pi),.$$