On transcendental progressions, that is, those whose general terms cannot be given algebraically
One of Euler’s earliest papers, he begins with Wallis's "hypergeometric series" 1! + 2! + 3! + 4! + ... (he didn’t have the "!" factorial notation yet) to find the value of the function for a general value of x. He defines [x] to be ∫01 (ln(1/t))x dt, which after substituting t = e-z gives [x] = ∫0∞ zx e-z dz, what we know as the Gamma function. He finds [1/2] to be √(π/2) and derives the recursive rule [x+1] = (x+1)∙[x]. He finds [p/q] to be a product of beta functions, and derives a differential quotient.
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 36-57.
Opera Omnia Citation
Series 1, Volume 14, pp.1-24.