## 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 $$V_n$$.

### MSC:

 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