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    Proving Factorial Equations involving Double Factorials

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    In many problems in mathematical physics, particularly in connection with Legendre polynomials (Chapter 12), we encounter products of the odd positive integers and products of the even positive integers. For convenience, these are given special labels as double factorials:

    1*3*5***(2n + 1) = (2n + 1)!!

    2*4*6***(2n) = (2n)!!

    Show that these are related to the regular factorial functions by

    (2n)!! = (2^n)n! and (2n + 1)!! = (2n + 1)!/[(2^n)(n!)].

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    Solution Summary

    The solution completes proofs of the relationships between the double factorials and the regular factorials are provided, including a detailed explanation of each.