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As far as I can tell, the first occurrence of the sum notation ($\Sigma$) was in Euler's book Institutiones calculi differentialis:
Quemadmodum ad differentiam denotandam usi sumus signo $\Delta$, ita summam indicabimus signo $\Sigma$...
(Just as we used the symbol $\Delta$ to signify a difference, so we use the symbol $\Sigma$ to indicate a sum...)
But I can't seem to find any indication of where the product symbol ($\Pi$) first occurred. Does anyone know the source?
Gullberg's Mathematics: From the Birth of Numbers attributes first use to Descartes. This is not exactly a history book, and it comes with no source reference, so it is not very credible. If Descartes did use it it would have been a historical accident with no connection to modern use. Vieta before him and Wallis after him, along with a host of 17-18th century authors including Euler, just wrote several factors followed by some equivalent of "etc.", a whole sentence for Vieta and just "&c." for Wallis. Modern use of ${\small \Pi}$ tracks to 19-th century.
Cajori's History of Mathematical Notations, v.2, p.78 cites Gauss's Disquisitiones Generales circa Seriem Infinitam as the earliest use, but what Gauss introduces there is the so-called Pi function, not the product symbol: $$ {\small \Pi (k,z)={\frac {~1~~~\,\cdot\,~~~2~~~\,\cdot\,~~~3\,\,.\,.\,.\,.\,.\,\cdot\,k~~~}{(z+1)(z+2)(z+3)\,.\,.\,.\,.\,(z+k)}}k^{z}\,.} $$ I suppose Cajori believed that this is from where the notation spread, eventually mutating in meaning. His reference to Jordan's Traite de Substitutionis (1870) is the earliest where the use is recognizably modern. There was a discussion of big Pi by Wikipedia editors one of whom pointed to earlier occurrences:
"Did a bit of digging myself, and found one use of $\Sigma$ for summation in Fourier's 1822 Théorie analytique de la chaleur and one in Cauchy's 1826 Leçons sur les applications de calcul infinitésimal. Both authors assume the reader is unfamiliar with the notation and provide an explanation. Some thirty years later, Riemann in his 1859 Über die Anzahl der Primzahlen unter einer gegebenen Größe and Dedekind in his 1863 Vorlesungen über Zahlentheorie are both using $\Sigma$ and $\Pi$ without further explanation. This indicates that the summation and product notations passed into general usage sometime between 1820 and 1860.".
It seems that one would have to pore over first half of 19th century papers and books to locate the "earliest" use.
According to Cajori's A History of Mathematical Notations Volume 2 the notation first appeared in Gauss' "Disquisitiones generales circa seriem infinitam $1+\frac{\alpha\beta}{1\cdot\gamma}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}xx+\frac{\alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2)}{1\cdot2\cdot3\cdot\gamma(\gamma+1)(\gamma+2)}x^3+$ etc" which appears in Werke Vol III.
