A unified matrix approach to the Legendre-Sheffer and certain hybrid polynomial sequences (Q6645304)

The two-variable Legendre polynomials \(S_n(x,y)\) are introduced using the generating function\N\[\Ne^{yt}C_0(-xt^2)=\sum_{n=0}^\infty S_n(x,y)\frac{t^n}{n!},\N\]\Nwhere \(C_0(xt)\) denotes the 0-th order Bessel-Tricomi function, \(C_0(\alpha x)=\exp(-\alpha D_x^{-1})\{1\}\).\N\NOn the other hand, the Sheffer sequence \(s_n(x)\) is uniquely determined by two (formal) power series: \N\[\Nf(t)=\sum_{n=0}^\infty f_n\frac{t^n}{n!}, \qquad f_0=0, f_1\not=0,\]\Nand\N\[\Ng(t)=\sum_{n=0}^\infty g_n\frac{t^n}{n!}, \qquad g_n\not=0.\N\]\NThen there exists a unique sequence of polynomials \(s_n(x)\) such that\N\[\N\langle g(t)f(t)^k | s_n(x)\rangle=n!\delta_{n,k}, \qquad \forall n,k\geq 0,\N\]\Nwhere the exponential generating function \(s_n(x)\) is given by \N\[\N\frac{e^{xf^{-1}(t)}}{g(f^{-1}(t))}=\sum_{n=0}^\infty s_n(x)\frac{t^n}{n!} \qquad \forall x\in\mathbb{C},\N\] \Nand \(f^{-1}(t)\) is the compositional inverse of \(f\). In this case, \(s_n(x)\) is the Sheffer sequence for the pair \((f(t),g(t))\).\N\NNow, the hybrid Legendre-Sheffer polynomials \({ }_{L_e}s_n(x,y)\) are defined as \N\[\N\frac{1}{g(f^{-1}(t))}e^{yf^{-1}(t)}C_0(-x(f^{-1}(t))^2)=\sum_{n=0}^\infty { }_{L_e}s_n(x,y)\frac{t^n}{n!}.\N\]\N\NIn this context the authors obtain recursive formulas for the Legendre-Sheffer polynomials by means of the properties between the Pascal functional and Wronskian matrices. For this purpose, they introduce the vector version of the Legendre-Sheffer polynomials:\N\[\N\overline{{ }_{L_e}s_n(x,y)}= [{ }_{L_e}s_0(x,y) { }_{L_e}s_1(x,y)\dots { }_{L_e}s_n(x,y)]^T,\N\]\Nand express \({ }_{L_e}s_{n*1}(x,y)\) in terms of \({ }_{L_e}s_n(x,y)\). Also, differential equations for the Legendre-Sheffer polynomials are established and certain examples of these polynomials are given.