Differential properties of a general class of polynomials. (English) Zbl 0844.11016

Let \(x\), \(p\), \(q\) be real numbers, and \(U_n= U_n (p, q, x)\); \(V_n= V_n (p, q, x)\) be defined by the recurrences \(U_n= (x+ p) U_{n-1}- qU_{n-2}\), \(V_n= (x+ p)V_{n-1}- qV_{n-2}\) \((n\geq 2)\); \(U_0 =0\), \(U_1= 1\); \(V_0= 2\), \(V_1= x+p\). The author considers certain differential properties of \(U_n\) and \(V_n\) as functions of \(x\). For example, the polynomials \(U_n^{(k- 1)}= (d^{k-1}/ dx^{k-1} )U_n\) \((k\geq 1)\) and \(V_n^{(k)}= (d^k/ dx^k) V_n\) \((k\geq 0)\) satisfy the differential equation \(Az''+ (2k+1) (x+p) z'+ (k^2- n^2) z=0\), where \(A= (x+ p)^2- 4q\). In two preliminary papers, the author has discussed combinatorial properties of these coefficients \(U_n\) and \(V_n\).


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
11B83 Special sequences and polynomials