Some properties of a class of polynomials (Q2757728)

The polynomials \(U_n(p,q;x)\), defined by the recurrence relation NEWLINE\[NEWLINEU_n(p,q;x)=(x+p)U_{n-1}(p,q;x)-qU_{n-2}(p,q;x),\quad n\geq 2,NEWLINE\]NEWLINE with \(U_0(p,q;x)=0\) and \(U_1(p,q;x)=1\), were studied by \textit{R. André-Jeannin} [Fibonacci Q. 32, 445-454 (1994; Zbl 0822.11018); ibid. 33, 341-351 (1995; Zbl 0832.11010)]. These polynomials include as special cases the Fibonacci polynomials \(F_n (x)\), the Pell polynomials \(P_n (x)\), the first Fermat polynomials \(\Phi_n(x)\), the Morgan-Voyce polynomials of the second type \(B_n (x)\), and the Chebyshev polynomials of the second type \(S_n (x)\).NEWLINENEWLINENEWLINEHere the author considers the polynomials \(U_{n,m}(p,q;x)\), given by NEWLINE\[NEWLINEU_{n,m}(p,q;x)=(x+p)U_{n-1,m}(p,q;x)-qU_{n-m,m}(p,q;x),\quad n\geq m,NEWLINE\]NEWLINE with starting polynomials \(U_{0,m}(p,q;x)=0\), \(U_{n,m}(p,q;x)=(x+p)^{n-1}\), \(n=1,2,\ldots,m-1\). She determines the coefficients \(c_{n,k}(p,q)\) of these polynomials and defines the polynomials \(f_{n,m}(p,q;x)\), which are the rising diagonal polynomials of \(U_{n,m}(p,q;x)\).