2
$\begingroup$

I want to know who was the first scholar or mathematician to have introduced and formulate the concept of recurrence relations, that is finding a function given the how one value of a sequence is related to the previous ones.

$\endgroup$
10
$\begingroup$

Depends on what "introduced" means. If we skip arithmetic and geometric series as too simple to suggest something general the idea goes back to Fibonacci and his rabbit breeding problem that sets up a recurrence in Liber Abaci (1202). Jacob (1564) and Kepler (1618) pointed out that the limit ratios of consecutive terms approach the golden ratio, see Tatlow, The Use and Abuse of Fibonacci Numbers for the early pre-history. Girard wrote down the recurrence explicitly in 1634, and De Moivre (c. 1718) and Bernoulli (1726) solved it explicitly, i.e. derived what is now called the Binet formula, see With what kind of proof was the Binet formula derived for the first time? But this was developing general ideas on a concrete example rather than introducing a more abstract concept explicitly.

De Moivre's treatise Miscellanea analytica de seriebus et quadraturis (1730) is sometimes credited as the first systematic study of linear recurrences. He also introduced the idea of a generating function, and derived it for the Fibonacci sequence. But in a different guise, as solving finite difference equations, a general approach appears already in Taylor's treatise Methodus Incrementorum (1715), see Feigenbaum, Brook Taylor and the Method of Increments. Propositions 6-9 cover some methods for solving difference equations, including by infinite series. Boole's Treatise on the Calculus of Finite Differences (1859) was a continuation of this, but like other literature on finite differences it did not focus on topics usually associated with recurrent sequences as such today.

Judging by the historical survey in Dickson's History of the Theory of Numbers, ch. XVII, many of the modern themes and techniques, especially number theoretic ones, were first put together and systematized in a series of works by Lucas in 1876-78, as applied to the second order sequences. They became the template for treating higher order sequences in later works. A little later, Poincaré initiated a general study of the asymptotics of solutions to difference equations in Sur les equations lineaires aux differentieles et aux difference finies (1885), see Abu-Saris et al., Poincaré Types Solutions of Systems of Difference Equations. In particular, he proved a first general result on the limit ratios of higher order sequences, which generalize the golden ratio.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.