Sigma function solution of the initial value problem for Somos 5 sequences

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve $X$, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of $X$. The construction is based on a Lax pair with a spectral curve $S$ of genus four admitting an involution $σ$ with two fixed points, and the Jacobian of $X$ arises as the Prym variety Prym$(S,σ)$.

Somos-4 and Somos-5 are arithmetic divisibility sequences

We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by $τ_{n+2}τ_{n-2}=ατ_{n+1}τ_{n-1}+βτ_n^2$ and initial values $τ_1=τ_2=τ_3=τ_4=1$, that the polynomial $τ_n(α,β)$ is a divisor of $τ_{n+k(2n-5)}(α,β)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.

Somos 4序列的整性性与Laurent现象

Integrality and the Laurent phenomenon for Somos 4 sequences

本文证明Somos 4序列满足强于Laurent性质的条件，建立整性性的充分条件，并给出非周期序列对应的四元四次丢番图方程的无穷多解，同时推广到Somos 5序列。

在阴影中投射更多光芒：对偶 Somos-5 序列

Casting more light in the shadows: dual Somos-5 sequences

本文研究对偶数环上最一般形式的 Somos-5 递推，给出其初值问题通解及阴影序列的三种表示：Weierstrass sigma 函数解析形式、线性差分方程解以及基于 Hankel 行列式与 QRT 映射的联系。

汉克尔变换与(α,β)型Somos-4序列

Hankel Transform and $(α,β)$ Somos-4 Sequences

本文在有理函数域扩展Wang-Zhang充分条件，结合Sulanke-Xin二次变换解决Barry的(α,β)型Somos-4相关猜想，证明该序列奇偶子序列性质及汉克尔行列式相关猜想。

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model $y^2=4x^5+c_4 x^4+ ... +c_1 x+c_0$. We show that the recurrence sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves…