Generalized order-\(k\) Fibonacci and Lucas functions
For any real number \(x\) and \(1\leq i\leq k\), the authors define and study the function \[ F(i,k,x)= \sum^\infty_{t=0} (g^i_t/t!)x^t, \] where \[ g^i_n= \sum^k_{j=1} g^i_{n-j}\quad (n> 0,\;1\leq i\leq k), \] and \(g^i_n= 1\) if \(i= 1-n\) and \(0\) otherwise \((1-k\leq n\leq 0)\). They name \(F(i,k,x)\) generalized order-\(k\) Fibonacci function. They also define and study a function \(L(i,k,x)\), which they call generalized order-\(k\) Lucas function.
generalized order-\(k\) Fibonacci and Lucas numbers
scientific article; zbMATH DE number 5541308