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We consider linear-fractional branching processes (one-type and two-type) with immigration in varying environments. For 
$n\ge0$, let 
$Z_n$ count the number of individuals of the nth generation, which excludes the immigrant who enters the system at time n. We call n a regeneration time if 
$Z_n=0$. For both the one-type and two-type cases, we give criteria for the finiteness or infiniteness of the number of regeneration times. We then construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. For example, it may happen that the process is extinct, but there are only finitely many regeneration times. We also study the asymptotics of the number of regeneration times of the model in the example.
$(\,j,k)$-ENTRY 
$q^{\,j\pm k}+t$
We determine the characteristic polynomials of the matrices 
$[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and 
$[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number 
$q\not =0,1$. As an application, for complex numbers 
$a,b,c$ with 
$b\not =0$ and 
$a^2\not =4b$, and the sequence 
$(w_m)_{m\in \mathbb Z}$ with 
$w_{m+1}=aw_m-bw_{m-1}$ for all 
$m\in \mathbb Z$, we determine the exact value of 
$\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.
