Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true 
$$F(a,b,c;x)=\sum\_{n=0}^{+\infty} \frac{(a)\_n(b)\_n}{(c)\_n(1)\_n} x^n=\frac{\Gamma ( c )}{\Gamma(a)\Gamma(c-a)}\int\_0^1 t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b} dt\, , $$
where $(d)\_n:=d(d+1)\cdots(d+n-1)$ for every $d$ and every $n\in \mathbb N$. 


To whom is due this formula (Euler?) and where **precisely** can I find its first appearance in the classical literature?


Thanks in advance.



First proof of the integral representation of the hypergeometric function $F(a,b,c;\cdot)$
The following paper discusses the hypergeometric functions and its history in mathematics (for the integral representation cf. p. 26 and references given there): 


* J. Dutka. The early history of the hypergeometric function. Archive for History of Exact Sciences Vol. 31, No. 1 (1984), pp. 15-34


[JSTOR link](http://www.jstor.org/stable/41133728)


According to Dutka, Euler came close to the integral representation in his investigation of the hypergeometric differential equation. Some integral representation was found by Legendre in 1816. Apparently the first appearance of the now-known integral representation is in the dissertation of P.C.O Vorsselman de Heer in 1833 (three years before Kummer). 


