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Nature of Legendre foretold by Pascal
P. R. Subramanian, B. V. Jenisha, and A. Jestin Lenus
Central Instrumentation and Service Laboratory, University of Madras, Guindy Campus, Chennai 600 025, India.
Each coefficient of a Legendre polynomial is a rational number. The product of each Legendre coefficient and the exact power of 2 which divides the factorial of the degree, is an integer. Each product is divisible by the the binomial coefficient associated with the degree of the polynomial and the corresponding index of the power of the argument. As their ratio is always an odd integer, each product and the associated binomial coefficient have the same parity. One of the interesting consequences of the divisibility property and the same parity is the production of a palindrome by the products with respect to their parity. The divisibility property can be extended to the Hermite and Laguerre polynomials.