Volterra-type convolution of classical polynomials
Provides new formulas for convolutions of classical orthogonal polynomials, which may be useful for researchers in special functions and applied mathematics, but the work is incremental.
The paper develops a general framework for computing Volterra-type convolutions of polynomial sequences and derives series representations for convolutions of classical orthogonal polynomials like Jacobi and Laguerre families.
We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $°P_k(x) = k$. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.