On polynomials associated with Humbert's polynomials (Q2774617)

The authors consider the polynomials \(\Theta_n(x)\equiv p_{n,m,a,b,c,d}^\nu(x)\) defined by the generating function NEWLINE\[NEWLINE \bigl(c-axt+bt^m(2x-1)^d\bigr)^{-\nu}=\sum_{n=0}^\infty \Theta_n(x)t^n. NEWLINE\]NEWLINE For \(a=m\), \(d=0\), \(\nu=-p\), these polynomials reduce to generalized Humbert polynomials, which were intensively studied by \textit{H. W. Gould} [Duke Math. J. 32, 697-711 (1965; Zbl 0135.12001)]. Some special cases were considered by \textit{A. F. Horadam} and \textit{S. Pethe} [Fibonacci Quart. 19, 393-398 (1981; Zbl 0477.10013)], \textit{A. F. Horadam} [Fibonacci Q. 23, 294-299, 307 (1985; Zbl 0589.10016)], \textit{K. Dilcher} [SIAM J. Math. Anal. 19, No. 2, 473-483 (1988; Zbl 0642.33014)], \textit{G. V. Milovanović} and \textit{G. B. Djordjević} [Fibonacci Q. 25, 356-360 (1987; Zbl 0635.33006)], [Facta Univ., Ser. Math. Inf. 6, 23-30 (1991; Zbl 0812.33004)], and \textit{S. K. Sinha} [Proc. Natl. Acad. Sci. India, Sect. A 59, No. 3, 439-455 (1989; Zbl 0725.33006)]. The authors give some finite series representations and a hypergeometric representation for \(\Theta_n(x)\), as well as an additional generating function for these polynomials. Also, they obtain certain expansions of \(\Theta_n(x)\) in series of classical (Legendre, Gegenbauer, Hermite, and Laguerre) orthogonal polynomials.