The Perrin Conjugate and the Laguerre Orthogonal Polynomial
The exponential expansion of the Perrin conjugate leads to a series like the exponential generating function for the Laguerre polynomial. This orthogonal polynomial can be used to expand any polynomial in a series of Laguerre polynomials. A summation series has been developed for the classic orthogonal polynomials. Integral representations are derived using the orthogonality of the Laguerre polynomial to find monomial terms of Legendre, Hermite and Chebyshev polynomials in terms of the Gamma function. Expansions can also be easily derived for these classical polynomials using the confluent hypergeometric function. The connection of these polynomials to symmetric functions is also demonstrated.