The two identities areTo prove them, expand in its exponential series. Since on , the Weierstrass M-test justifies termwise integration. Substituting gives through the gamma function. Inserting these moments into the series and shifting proves both formulas. Each resulting numerical series has absolute convergence.
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The "Sophomore's Dream" is a term used in mathematics, particularly in the context of number theory. It refers to a specific type of mathematical problem or equation related to the sums of squares and their properties. More specifically, it describes the scenario where a number can be expressed as the sum of two squares in more than one way.