Proofs for certain Pi_{q}-conjectures of Gosper

Proofs for certain conjectures of Gosper on q-trigonometric identities

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.

Proofs for two Lambert series identities of Gosper

Applying the theory of modular forms and Lambert series manipulations we establish an Eisenstein series identity. From this formula we confirm a Lambert series identity conjectured by Gosper. Another Lambert series identity of Gosper is also confirmed by using Lambert series manipulations.

Modular Proofs of Gosper's Identities

We give unified modular proofs to all of Gosper's identities on the $q$-constant $Π_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,\cdots,n_m$ with $m\geq 3$, $Π_{q^{n_1}}$, $\cdots$, $Π_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on $Π_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gosper's identities using hauptmoduls for genus zero congruence subgroups.

Gosper 超几何级数奇异求值的 q-模拟

A $q$-analogue of Gosper's strange evaluation of the hypergeometric series

本文利用基本超几何级数三项关系，构造并推广了 Gosper 提出的含两个自由参数的 ${}_{2}F_{1}$ 级数恒等式的 q-模拟。

The proof of conjecture of Brutman and Passow

We consider the conjecture of Brutman and Pasow on a totality divided differences and prove the conjecture for continuous functions.

The Proof of Alzer's Conjecture on Generalized Logarithmic Mean

In 1987, Alzer posed a conjecture on generalized logarithmic mean, which was introduced by Stolarsky in 1975. To prove Alzer's conjecture, Lou posed a conjecture on generalized inverse harmonic mean in 1995. By proving Lou's conjecture, the paper yields Alzer's conjecture finally.