The derivative connecting problems for some classical polynomials

Abstract

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ Let $\{ S_n(x) \}_{n\geq 0}$ be another polynomial set with $\deg ( S_n(x) )=n.$ The general connection problem between them consists in finding the coefficients $\alpha^{(n)}_{i,j}$ in the expansion $$Q_n(x) =\sum_{i,j=0}^{n} \alpha^{(n)}_{i,j} P_i(x) S_{j}(x).$$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=P'_{n+1}(x)$ the connection problem is called the derivative connecting problem and the general derivative connecting problem associated to $\{ P_n(x) \}_{n\geq 0}.$
 In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials.