\(h\)-analogue of Fibonacci numbers
Let \[ (a)_{s:n} = \begin{cases} 1, & n = 0 \\ a(a+s) \dots a(a+(n-1)s), & n = 1,2, \dots \end{cases} \] For \(s=1\), \((a)_{1:k}=(a)_{k}\) coincides with Pochhammer's symbol. Let \[ \begin{bmatrix} n \\ k \end{bmatrix}_{h,h'} =\begin{pmatrix} n \\ k \end{pmatrix}(hh')_{n;k}. \] The \(h\)-Fibonacci number is defined for \(n > 0\) by \[ F_n^{(h,h')} = \sum_{k=0}^{\left [ \frac{n-1}{2} \right ]} \begin{bmatrix} n - 1 - k \\ k \end{bmatrix}_{h,h'}. \] Some properties of these numbers are formulated and proved, e.g. \[ F_n^{(h,h')} = _3F_1 \sum_{k=0}^{\left [ \frac{n-1}{2} \right ]} \frac{(-\frac{n}{2} + \frac{1}{2})_k(-\frac{n}{2} + 1)_k(h')_k}{(-n+1)_k}. \frac{(-4h)^k}{k!}, \] \[ F_{n+1}^{(h,h')} = F_n^{(h,h')} + hh'F_{n-1}^{(h,h'+1)}, \] \[ F_{n+2}^{(h,h')} = hh'\sum_{k=1}^n F_k^{(h,h'+1)} + 1, \] and others.
generalized Fibonacci numbers and polynomials
scientific article; zbMATH DE number 6358123