The derivative connecting problems for some classical polynomials

A. Ramskyi, N. Samaruk, O. Poplavska

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 derivative 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 stillof interest. The connection coefficients play an important role in many problems inpure and applied mathematics, especially in combinatorics,   mathematicalphysics  and quantum chemical applications.For the particular case $Q_n(x)=P'_{n+1}(x)$  and  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.

Keywords


connection problem; inversion problem; derivative connecting problem; connecting coefficients; orthogonal polynomials

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