A Discrete Analogue for Jacobi Polynomials
Start Date
9-4-2024 12:00 AM
Description
Classical Jacobi polynomials are a family of orthogonal polynomial sequences that appear in numerous physical sciences. We investigate discrete analogues of Jacobi polynomials through their hypergeometric representation, resulting in "discrete Jacobi polynomials" that satisfy qualitatively similar properties to their continuous counterparts. A difference equation analogue of the classical Jacobi polynomial differential equation and an analogue of its three-term recurrence are derived. This expands the growing subject of discrete analogues of special functions. A crucial part of this work is to generalize the theory of discrete analogues of generalized hypergeometric functions: in the current literature, finding analogues with monomial arguments of the classical hypergeometric has been solved, but we instead consider a linear argument for which existing methods fail. Our new technique can be applied to the existing discrete Legendre and discrete Chebyshev polynomials, which previously used a workaround that altered the final resulting formulas in an unsatisfying way.
Recommended Citation
Barnes, Drew, "A Discrete Analogue for Jacobi Polynomials" (2024). 2024 Student Academic Showcase. 13.
https://digitalcommons.lindenwood.edu/src_2024/Posters/Session1/13
A Discrete Analogue for Jacobi Polynomials
Classical Jacobi polynomials are a family of orthogonal polynomial sequences that appear in numerous physical sciences. We investigate discrete analogues of Jacobi polynomials through their hypergeometric representation, resulting in "discrete Jacobi polynomials" that satisfy qualitatively similar properties to their continuous counterparts. A difference equation analogue of the classical Jacobi polynomial differential equation and an analogue of its three-term recurrence are derived. This expands the growing subject of discrete analogues of special functions. A crucial part of this work is to generalize the theory of discrete analogues of generalized hypergeometric functions: in the current literature, finding analogues with monomial arguments of the classical hypergeometric has been solved, but we instead consider a linear argument for which existing methods fail. Our new technique can be applied to the existing discrete Legendre and discrete Chebyshev polynomials, which previously used a workaround that altered the final resulting formulas in an unsatisfying way.