The derivative connecting problems for some classical polynomials - Ramskyi, A.; Samaruk, N.; Poplavska, O. (Carpathian Mathematical Publications, 2019)
Given two polynomial sets { P_n(x) }_n≥ 0, and { Q_n(x) }_n≥ 0 such that ( P_n(x) ) =n, ( Q_n(x) )=n. The so-called the connecting problem between them asks to find the coefficients α_n,k in the expression Q_n(x) =_k=0^n α_n,k P_k(x). Let { S_n(x) }_n≥ 0 be another polynomial set with ( S_n(x) )=n. The general connection problem between them consists in finding the coefficients α^(n)_i,j in the expansion Q_n(x) =_i,j=0^n α^(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≥ 0. In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials.