In mathematics, in combinatorics, the Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Shanlan.
[2] This identity appears in the third chapter of Duoji bilei (垛积比类 / 垛積比類, meaning summing finite series), a mathematical text authored by Li Shanlan and published in 1867 as part of his collected works.
[3] Kaucky attributed the identity to a certain Li Jen-Shu.
[4] Li Shanlan had not given a proof of the identity in Duoji bilei.
The first proof using differential equations and Legendre polynomials, concepts foreign to Li, was published by Pál Turán in 1936, and the proof appeared in Chinese in Yung Chang's paper published in 1939.
is independent of k, this can be put in the form Next, the result gives Setting p = q and replacing j by k, Li's identity follows from this by replacing n by n + p and doing some rearrangement of terms in the resulting expression: The term duoji denotes a certain traditional Chinese method of computing sums of piles.
Most of the mathematics that was developed in China since the sixteenth century is related to the duoji method.