Integral transforms and Chebyshev-like polynomials.
The author considers the Chebyshev polynomials \[ U_n(x, y)=\sum_{k=0}^{[n/2]}\frac{(n-k)! y^k x^{n-2k}}{k!(n-2k)!} \] and their \(m\)th-order generalized forms \[ U_n^{(m)}(x, y)=\sum_{k=0}^{[n/m]}\frac{(n-k)! y^k x^{n-mk}}{k!(n-mk)!} \] Some properties of these polynomials are obtained by means of the integral representation method. For instance, it was shown that the equation \[ \left\lfloor (4y+x^2)\;\frac{\partial^2}{\partial x^2} + 3x\frac{\partial}{\partial x} -n(n+2)\right\rfloor U_n(x, y)=0, \] and the recurrence \[ U_{n+1}^{(m)} (x, y)=x U_n^{(m)}(x, y) +(m-1)y \frac{\partial^{m-2}}{\partial x^{m-2}} U_{n-1}^{(m)} (x, y) \] hold. Other families of polynomials are also considered.
scientific article; zbMATH DE number 2012690