A family of continued fractions
The author investigates a one-parameter family of infinite generalized continued fractions \[ \frac{1+s}{1+}\frac{2+s}{2+}\frac{3+s}{3+}\cdots \] He proves that for any positive integer \(s\) this continued fraction converges to a rational number \[ \frac{w(0,s)}{w(1,s)}, \] where \(w(z,s)\) is a sequence of polynomials defined by formulae \[ w(z,1)=1,\qquad w(z,2)=z+4, \] \[ w(z,s)=(z+2s)w(z,s-1)-(s-2)(z+s)w(z,s-2)\quad\text{for}\quad s\geq 3. \] In the second part of the paper the author describes interesting combinatorial properties of the numerators \(u(s)=w(0,s)\) and denominators \(v(s)=w(-1,s)\). Other properties of these numbers can be found in the Online Encyclopedia of Integer Sequences (see sequences A052852 and A000262) [see Zbl 0845.11001].
scientific article; zbMATH DE number 5690726