We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.
We determine the characteristic polynomials of the matrices 
$[q^{,j-k}+t]_{1le ,j,kle n}$ and 
$[q^{,j+k}+t]_{1le ,j,kle n}$ for any complex number 
$qnot =0,1$. As an application, for complex numbers 
$a,b,c$ with 
$bnot =0$ and 
$a^2not =4b$, and the sequence 
$(w_m)_{min mathbb Z}$ with 
$w_{m+1}=aw_m-bw_{m-1}$ for all 
$min mathbb Z$, we determine the exact value of 
我们确定了任意复数 $qnot =0,1$ 的矩阵 $[q^{,j-k}+t]_{1le,j,kle n}$ 和 $[q^{,j+k}+t]_{1le,j,kle n}$ 的特征多项式。作为应用，对于复数 $a,b,c$，其中 $bnot =0$ 和 $a^2not=4b$，以及序列 $(w_m)_{min mathbb Z}$，其中对于所有 $min mathbb Z$，$w_{m+1}=aw_m-bw_{m-1}$、我们确定 $det [w_{,j-k}+cdelta _{jk}]_{1le ,j,kle n}$ 的精确值。
An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith. 199 (2021), 33–53] introduced an analogue 
$overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for 
$overline {a}_t(n)$ where 
$t in {3,4,5,8}$ and studied the arithmetic density of 
$overline {a}_t(n)$ modulo 
$p_i^{,j}$ where 
$t=p_1^{a_1}cdots p_m^{a_m}$ and 
$p_igeq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq. 27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by 
Gireesh 等人['A new analogue of t-core partitions', Acta Arith.他们得到了 $overline {a}_t(n)$ 的乘法公式和算术等式，其中 $t in {3,4,5,8}$ 并研究了 $overline {a}_t(n)$ modulo $p_i^{,j}$ 的算术密度，其中 $t=p_1^{a_1}cdots p_m^{a_m}$ 和 $p_igeq 5$ 都是素数。Bandyopadhyay 和 Baruah [' Arithmetic identities for some analogs of the 5-core partition function', J. Integer Seq.27 (2024), 文章编号 24.4.5]证明了 $overline {a}_5(n)$ 所满足的新算术等式。我们研究了 $/overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^alpha m$ 的算术密度，其中 $gcd (m,6)$=1.另外，利用小野和田口的一个结果['某些模块形式的 2-adic 属性及其在算术函数中的应用'，Int.J. Number Theory 1 (2005), 75-101]关于赫克算子零点性的结果，我们证明了 $overline {a}_3(n)$ modulo arbitrary powers of 2 的无穷同余族。
We consider the existence problem of meromorphic solutions of the Fermat-type difference equation 
$$ begin{align*} f(z)^p+f(z+c)^q=h(z), end{align*} $$
where 
$p,q$ are positive integers, and h has few zeros and poles in the sense that 
$N(r,h) + N(r,1/h) = S(r,h)$. As a particular case, we consider 
$h=e^g$, where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense 
$T(r,h)=S(r,f)$.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：