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As far as I understand, the Gamma function was developped as a way of calculating "the" factorial of a non-integer number. Why did this problem interest 18th century mathematicians? Was it just a puzzle, or did they have some specific application in mind?

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The problem can be seen as part of more general attempts to extend the domain of operations defined initially for naturals (integers, or rationals), only. A prominent natural example would be taking an $n$-th power (another would be the binomial coefficients, and related Newtonian series). The problem is, thus, not an isolated one, but rather part of a general and natural project of that time.

An application that directly motivated the invention of the Gamma function was Goldbach trying to find something like a closed form for $\sum_{k=1}^n k!$.

This answer is based on the information given in Section 1 of:

Gronau, D. Why is the gamma function so as it is. Teaching Mathematics and Computer Science, 1, 43-53 (2003).

The reference mentioned there to support this point of view specifically is :

J. Christoph Scriba, "Von Pascals Dreieck zu Eulers Gamma-Funktion, Zur Entwicklung der Methodik der Interpolation", in: Mathematical Perspectives, Essays on Mathematics and Its Historical Development, (Joseph W. D auben, ed.), Academic Press, New York, 1981

I could not read this reference, but even only the title (my translation) "From Pascal's triangle to Euler's Gamma function, On the developpment of the method of interpolation." supports it.

The MO-question Who invented the gamma function? contains related information.

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Euler's motivation

R. Hilfer on pg. 18 of "Threefold Introduction to Fractional Derivatives" states, "Derivatives of non-integer (fractional) order motivated Euler to introduce the Gamma function ...." Euler introduced in the same reference given by Hilfer essentially

$$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$$

Related MSE-Q&A.

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  • $\begingroup$ See also "Construction and physical application of the fractional calculus" by N. Wheeler $\endgroup$ – Tom Copeland Jun 9 '15 at 18:00
  • $\begingroup$ See also "Fractional derivatives and special functions" by Lavoie, Osler, and Trembley. $\endgroup$ – Tom Copeland Jan 1 '16 at 17:07
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The reasons are the same as extending the binomial formula to non-positive exponents (Newton binomial). It involves generalized binomial coefficients which cannot be expressed in factorials. Binomial formula with non-integer exponent arises, for example in the computation of the arc length of an ellipse, and in other natural problems.

The first part of the survey by J. Lagarias, Euler's constant, in Bull AMS, 50 (2013) 527-628 (freely available online) contains a detailed analysis of Euler's work, and in particular addresses the need to interpolate various functions of integers with analytic functions.

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Interpolation became a big thing after Wallis's Algebra, where he first interpolated various sequences just for the fun of it. Since Wallis's "proofs" were hardly more than a sequence of educated guesses, Goldbach and Euler used Wallis's work as a collection of open problems.

Later, Euler would use the same methods for extending functions such as the zeta function to negative numbers and could even deduce (conjecturally) the functional equation of zeta(s) as well as for the Dirichlet L-function for the Dirichlet character $(-1/p)$ with conductor 4.

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